What is an event in Special Relativity? Lorentz transformations help us transform coordinates of one frame to that of another.
For example, let the coordinates of an event in an inertial frame $S$ be $(x, t)$, then the coordinates in frame $S'$ is got as $(x', t')$ using Lorentz transformations.
This means that there is some kind of equivalence between $(x, t)$ and $(x', t')$. But what kind of equivalence is it?
What I am trying to ask is What does it mean to say coordinates of the same event? What is an event?
Edit: I am probably wrong about this, but anyways: I think all the current answers are circular. I am asking about events so as to understand what spacetime means. But, all the answers involve spacetime in one way or the other. You can't just say an event is a location and a time because that is what I am trying to understand. Without absolute space and absolute time, what does that mean?
 A: In the original usage, as Einstein used it, an "event" is just something that happens, like a detector clicking. It's just the same as the colloquial meaning. 
In the early 20th century, various thought experiments involving hypothetical events, and actual experiments involving physical events, were used to show that general relativity is an excellent model for our universe. In the context of general relativity, spacetime is modeled as a Lorentzian manifold, and physical events are modeled as points in this manifold. 
Now, some mathematically minded people choose to forget all this history. They instead say the word "event" is defined as a point in a Lorentzian manifold. This is a clean and consistent definition, but as usual in mathematical physics, it misses the point. The only reason we care about these mathematical "events" is because they form part of a theory that does an excellent job of describing real events. By conflating the two, one glosses over the mountains of experimental work needed to link the two together. 
So you are completely right to notice that something fishy is going on. This linguistic bait and switch occurs constantly in physics courses. Typically one starts a course by defining spacetime and event in the usual, colloquial sense, but then ends the course by saying that spacetime is a Lorentzian manifold, that an event is a point. These are extremely different meanings of the same word, which are both commonly used, and it's important not to conflate them. The gap between them can only be bridged by experiment.
A: An event is any physical occurrence that we can consider to happen at a definite point in
space and at a definite
instant in time.  These could be approximated to the location of the nearest intersection of imaginary grid points in space, each carrying a synchronized clock. There is a famous picture of such a reference frame in the book by Taylor, Edwin F., and John Archibald Wheeler. Spacetime physics. Macmillan, 1992.

The numerical values $(x,y,z,t)$ 
uniquely locating an event depend on a certain number of arbitrary choices, such as the
location of the origin of the lattice and the moment considered to be $t=0$. 
A: Event can refer to real events which are for example:
- a firecracker exploding
- two objects scattering
- a photon getting emmited from an atom
Events also.dont have to.be real, it is enough (of course) that you could imagine e.g. a firecracker beiing exploded.
More abstractly an event is a point in space and time to which you can refer to.without using coordinates in space and time.
A: As others have stated, an event is a point of spacetime.  That's a good enough definition only if you understand what it means to be 'a point' and what 'spacetime' is: here is a lightning description of how that works.  This is not a complete description (or even, probably, correct in some places): I've added a couple of references at the end (which themselves are far from complete, they're just books I happen to have by me).
This has turned into a long answer: I hope it's still useful.
Topological spaces
So, you start of with a set, $X$ of things we'll call 'points': this set is usually infinite, and in fact uncountable, but it doesn't have to be (yet, it will below).
Now we want to set up some relations between points in $X$, which we do by defining a topology on $X$.  So, consider a collection of subsets of $X$, which I'll call $U$ (note: I'm not sure if $U$ is a set: I think you run into the standard Russell awfulness here and it might not be: that's why I'm calling it a 'collection').  $U$ must be such that:


*

*$X$ is in $U$ as is $\emptyset$;

*the union of every subcollection (see above) of $U$ is in $U$;

*the intersection of any finite number of subcollections of $U$ is in $U$.


The tuple $(X, U)$ then defines a topological space, and elements of $U$ are the open sets of the topological space.
I will give one example of a well-behaved topological space, which is the usual topology on $\mathbb{R}$.  Here, $X = \mathbb{R}$ and the points of $X$ are just real numbers.  We can then define $U$ as consisting of all open intervals, $(a, b), a, b \in \mathbb{R}, a < b$, and all the unions of such sets, with $\emptyset$ added.
It is fairly easy to check that $(\mathbb{R}, U)$ satisfies the topological axioms above.  What is more interesting is to see that, if you allow infinite intersections, things fall apart.  To do that consider an infinite intersection of open intervals $(p - 1/n, p + 1/n), n\in\mathbb{N}, p\in\mathbb{R}$: this is the point $p$ (this is easy to see as $p$ is the only point which belongs to all these sets), and yet $p$ is not the union of any collection of open intervals: in the usual topology you want points to be closed, not open.
There are other topologies, including other possible topologies for $\mathbb{R}$: two such are the topology containing only $\emptyset$ and $\mathbb{R}$, which is the trivial topology and the one where all subsets of $\mathbb{R}$ are in $U$, which is the discrete topology.  These are not interesting for our purposes other than to understand that you can choose your topology.
A neighbourhood of a point $p\in X$ is any subset of $X$ which contains an open set which contains $p$.  You need this two-level definition because you don't want to insist neighbourhoods are open.  An open neighbourhood is a neighbourhood which is also an open set.
There are a bunch of other important things about topologies which I'm just going to skip as I have no space or time, but they include things like the definitions of closed set, compactness, separability and a number of other really important things.
Continuity
A very important thing that you get once you have a topology is a notion of continuity. I'll assume you're happy with the idea of a mapping between two sets, and notions like whether a mapping is one-to-one &c.  We can define a mapping $f: M\to N$ (where $M$ and $N$ are topological spaces) as being continuous at some point $p\in N$, if any open set of $N$ containing $f(p)$ contains the image of an open set of $M$ under $f$.  $f$ is then continuous on $M$ if it is continuous at all points of $M$.
This definition of continuity is equivalent to the normal one for $\mathbb{R}$ if you assume the usual topology.  The normal definition of continuity is that $f:\mathbb{R}\to\mathbb{R}$ is continuous as $x\in\mathbb{R}$ if for every $\epsilon > 0$ there is a $\delta > 0$ such that $|f(y) - f(x)| < \epsilon$ whenever $|y - x| < \delta$.  But $(x - \delta, x + \delta)$ is an open set, as is $(f(x) - \epsilon, f(x) + \epsilon)$, and the second set is an open set containing $f(x)$, and also containing the image of the first, and any open set containing $f(x)$ will contain the image of an open set containing $x$ as we can make $\epsilon$ and $\delta$ as small as we like.
So the definitions of continuity are equivalent, but the topological one is much more general, because it does not rely on any notion of distance.
Manifolds
So, we've got points and a notion of topology and continuity, but we have not really tied things down very far, as we could have really odd topologies.  What we want to do is to define some kind of structure which is 'like' $\mathbb{R}^n$, at least locally.  And that's what a manifold is.
A manifold is a topological space, $M$, where each point $p\in M$ has an open neighbourhood which has a continuous, one-to-one map onto an open subset of $\mathbb{R}^n$ for some $n$.  (It's safe to assume the usual topology on $\mathbb{R}^n$ I think: you could have manifolds where the topology on $\mathbb{R}^n$ was not the usual one but they'd be strange things.)  Note that the mappings only cover neighbourhoods: there's no need for there to be some global mapping, and in general there will not be (for instance, the surface of a sphere has no global one-to-one mapping to $\mathbb{R}^2$).  The elements of $\mathbb{R}^n$ in a mapping are the coordinates of the point $p\in M$ (and, obviously, there may be several such mappings for a given point $p\in M$ which you can construct just by considering mappings from $\mathbb{R}^n$ to $\mathbb{R}^n$).  This is the point at which we have to assume that there are uncountably many points, since we need there to be one-to-one mappings onto a set we know to be uncountable.
And now we can do a wonderful thing: we can use the whole mechanism of analysis on $\mathbb{R}^n$ to boostrap things like a notion of differentiability on the manifold.  I'll just give one definiton here and then stop.
If you think a bit you will realise that open sets are either disjoint or they have overlaps which are themselves open sets: they can't just touch at a single point.  It's easy to see this if you consider open intervals on $\mathbb{R}$: $(a, b)$ and $(c, d)$ are either disjoint (if $c \ge b$) or they have an overlap (if $c < b$).  This means that the mappings between $M$ and $\mathbb{R}^n$ must overlap.  So if we consider two such mappings from $M$ into $\mathbb{R^n}$ $f_1$ & $f_2$, then we can construct a mapping on the overlap $f(x) = f_2(f_1^{-1}(x))$, where $x$ is in the overlap.  This is a function from (an open subset of) $\mathbb{R}^n$ to $\mathbb{R}^n$, and we can ask questions about it: is it continuous (yes)?  Is it differentiable (not necessarily), and if it is how differentiable is it?
Well, a manifold where all these overlap mappings are differentiable is a differentiable manifold, and it's these things that form the basis of how relativity thinks about spacetime: spacetime is a manifold (with some additional structure) and events are points in it.

References


*

*Geometrical methods of mathematical physics by Bernard Schutz is a good starting place.

*Analysis, manifolds and physics by Y Choquet-Bruhat, C Dewitt-Morette with M Dillard-Bleick is a much more serious book.  I believe it may now exist as several smaller books, or alternatively have grown into many books: mine dates from 1985.

A: An event is just a specific point in spacetime, i.e., a specific combination of location and time.
In problems, there is often something physical that happens at a particular event, which provides one way of identifying which event you're talking about. For example, perhaps the event is where the front of a train car is, at the moment when a bolt of lightning strikes it. If there isn't something notable that happens at the event, it may be helpful conceptually to think of the event as if something notable does happen then and there, even if there isn't. For example, you could picture event "A" as being where a firecracker labeled "A" is, at the moment that it explodes.
Sometimes people get confused by special relativity because it seems like it involves two observers who disagree about what happens. For example, different observers will have different ways of describing where something is, or when something happens. But events are something useful to focus on because they are something that all observers can agree on, in that everybody agrees as to what events exist. For example, everybody can agree that there is a specific time and location at which the firecracker labeled "A" explodes. You could identify the event by calling it "the time and location at which firecracker A explodes", and everybody would agree as to which event is being talked about. Instead of using verbose names like that, it's more systematic and useful to label each event by assigning a set of four numbers to the event. There are different ways of assigning a set of four numbers to each event, but that's just a difference in naming systems, not a disagreement about what events exist.   
A: An event in spacetime is anythimg you deem important to mark by time and position (or 4D-spacetime location).
It can be the birth of your child, a collision of two particles, the death of Julius Cesar,...
A: I can easily understand the dissatisfaction for the answers. I am not sure if the mine would be better, but I can add something which I do not see has been discussed clearly before.
I agree that defining an event as a point in space-time can be highly non convincing, from the physical point of view. More mathematically oriented people could find it a good definition, but that's true only if one looks at the space-time of Relativity just as a mathematical structure. 
Luckily (for us who are inside), space-time is a concept aiming to describe the physical world we have experience and not the ideal constructions of our mind.
The situation is quite similar to the case of a related question: "what is a point in the space?". Difficulties with an answer for events in space-time closely  mirror difficulties in separating mathematical geometry and physical geometry. The two concepts are related but not overlapping.
Mathematical geometry (better, geometries) are intended to provide a mathematical model (models)  for something existing in the world. Once we are equipped with a good knowledge about mathematical geometries, we have the problem of identifying which among them is the most adapted to provide a faithful model of what exists in the real word. 
One first step toward this identification is to establish a correspondence between the undefined primitive elements of the geometries (point, line, plane, etc.) and something operationally accessible in the real world: objects, parts of objects, signs drawn on surfaces, beams of light,...). The exact way this correspondence is established is somewhat conventional. But, once this has been done, one can check (doing experiments and measurements) which of the possible geometric axioms are satisfied by the chosen set of primitive elements.
The delicate point in this procedure is precisely the very first one: the identification of the physical (measurable) entities with ideal entities. To make the whole situation worse, there is a different conceptual problem  overlapping with the problem of identifying the primitive entities. It is the problem if one looks at the physical geometry as a set of relations between "points" uniquely identified by a physical entity, or if one assigns some existence, even virtual, to geometric points, even without a coinciding physical entity. At the best of my knowledge, these two points of view are equally defensible from a philosophical point of view, and may coexist in practice, since their difference is not physically measurable.
After this quite long introduction, the question about events could have  relatively quick answer:
Events are physical points which have to be operationally identified and made corresponding to the geometrical description of an axiomatic structure which could be used to model the physical space-time.
How do we chose a candidate for an event? Our intuition says that the basic information brought about in the definition of an event should be something enabling us to establish spacial and temporal relations with other entities of the same kind. Even without a precise definition, we know how to do: we have to chose an identifiable change of a physical system. We need a physical system (not too much extended) to allow for measuring spacial relations with other systems. And we need a change to identify a "when". These considerations should make clear, why the emission of a photons, the crackling of a firework, a bullet hitting a target, are all simple example of what an event is: something (physical) which happens (in a reduced space and in a reduced time).

(Note added a few hours after)
A reasonably short definition and coherent definition of an event could be "A  phenomenon involving some change of a  physical system, which could be uniquely identified among many other phenomena of the same kind". Notice that such a definition does not involve an explicit reference to the space-time structure, but provides an operational way to identify entities on which to build the geometry of space-time.
A: Suppose your friends want to hang out. The agreement is: everyone meets at the bar at 7:00PM. 
Your friends want to invite you. If they just told you "hey, we'll meet at the bar", would you be able to show up? Certainly not, since you need to know when the meeting will occur.
What if they just told you: "hey, we'll meet at 7:00PM"? You still won't be able to show up, since you don't know where the meeting will occur.
It is clear then that for you to show up they need to give you full information on where and when the event (the meeting) will happen. Giving just the place or just the time won't work.
Similarly, Special Relativity cares a lot about when and where things happen. If we imagine a 3D space (with coordinates $x$, $y$ and $z$), we can certainly say where is a certain location (or point) by simply giving the $x$, $y$ and $z$ coordinates of that location.
We can also add a time coordinate, $t$, which we can use to specifiy when things happen. In the case of your friends it would be:
"Hey, the meeting will happen at coordinates $x$, $y$, $z$, at time $t$". Now you know when and where the meeting occurs, and you can easily show up.
We could then add a time coordinate to a 3D Space to create a 4D Spacetime. Now you've got four dimensions: 3 for space, 1 for time. A given point ($x$, $y$, $z$, $t$) in this 4D Spacetime IS an event. If another group of friends decided to meet at the bar earlier (say 6:00PM), would that be the same event as your meeting? No. The when is different: the $t$ coordinate is different.
Similarly, another group of friends could meet at the park at 7:00PM and you still have a different event: they were simultaneous, but the where is different.
All and all, we can conclude that an event can be described by where and when it happened, which is why we define that it is a point in spacetime. 
A: This is an answer specifically to the edit:

I think all the current answers are circular. I am asking about events so as to understand what spacetime means. But, all the answers involve spacetime in one way or the other. You can't just say an event is a location and a time because that is what I am trying to understand. Without absolute space and absolute time, what does that mean?

The problem that you are having is that the concept of an event is actually undefined. The “definition” of an event as a point in spacetime is intended to convey the notion of geometry.
The first relevant geometry was Euclidean geometry. In Euclidean geometry a point is an undefined concept, sometimes called a primitive. Euclid gave some illustrative examples to convey the concept of a point without defining it, and then simply proceeded to describe how they behaved using his famous axioms. This approach, of not defining the primitives but simply listing axioms of how they behave, has become common in all branches of math.
Next is Riemannian geometry. In Riemannian geometry the concepts of Euclidean geometry are generalized to describe geometry in curved surfaces where some of Euclid’s axioms don’t work. Points are still an undefined primitive notion, and now Riemann introduces the concept of a manifold which is a set of points with specific topological and geometric properties. Basically it is the space in which the geometry is being described. The topological properties of the manifold are described in terms of neighborhoods of points, and the geometric properties are defined in terms of the metric which describes the distance between neighboring points in the manifold. One of the key descriptors of a manifold is its signature which describes the dimension of the manifold. In a small neighborhood, a manifold with a signature (+++) can use the standard Pythagorean theorem to calculate distances, so the metric can be locally written $ds^2=dx^2+dy^2+dz^2$.
This brings us to the final relevant geometry which is pseudo-Riemannian geometry. The difference between Riemannian geometry is that now the signature is allowed to have negative as well as positive elements. So, a signature of (-+++) would mean that the metric could be locally written as $ds^2=-dt^2+dx^2+dy^2+dz^2$. When the signature has a single negative the geometry is often called Lorentzian instead of pseudo-Riemannian. This geometry still inherits the concepts of the manifold, the metric, and points, of which the point remains an undefined primitive. However, Lorentzian geometry (due to its usual application in physics) frequently renamed these concepts. The manifold is called spacetime and the points are called events, but they are still just the standard pseudo-Riemannian concepts of manifolds and points.
So, returning from the brief history to your question, in all of these geometries “point” is an undefined primitive. However, assuming that you already understand the concept of a point then you also understand the concept of an event. An event is simply a point in a (-+++) manifold which is called spacetime.
Now, you were trying to understand events in order to understand spacetime. But because events (points) are undefined primitives that may not be a viable approach, you will want to learn about manifolds on their own.
Let’s consider the one of the simplest manifolds, $R^2$, the flat 2 dimensional plane. $R^2$ can be seen as a collection of points, but in addition to simply being a set of points it has some additional structure. The first property is that for each point in $R^2$ there is a set of neighborhoods around that point, and those neighborhoods satisfy some axioms from topology that allow definitions of continuity and connectedness. This allows us to have things like paths and regions which are continuously connected sets of points. The next thing that $R^2$ has is a metric. The metric defines the length of any path. Once you have defined the metric then you can define things like angles and straight lines which are the shortest distance path between two points. All of the axioms of Euclidean geometry follow from the metric.
Now, we might find it convenient to add a coordinate system on top of $R^2$. If we do that then we can identify any point by its coordinates $(x,y)$ and we can write the metric in terms of the coordinates $ds^2=dx^2+dy^2$. We might find it so convenient to do so that we might exclusively identify points in that fashion, however, despite the convenience it is important to understand that the concepts of the manifold and the metric are geometric concepts that are independent of and more fundamental than the coordinates. The length of a path is a geometric quantity and it has a definite value regardless of the coordinate system we might use. We can change coordinate systems as much as we like, including rotations, translations, and even use all sorts of non-linear coordinates like polar coordinates. None of that changes the underlying geometry.
Hopefully all of that makes sense with respect to $R^2$. If so then we are nearly to the point where you can understand spacetime. Let’s consider a simple flat 2D spacetime. This is just like $R^2$ with one exception. Now, the metric is no longer positive definite, now the metric can be written $ds^2=-dt^2+dx^2$. This gives three distinct types of distances, distances where $ds^2<0$ called “timelike”, distances where $ds^2>0$ called “spacelike”, and distances where $ds^2=0$ called “null”. With this in place we are ready to map the geometrical ideas to physics. Spacelike distances are physically measured by rulers and timelike distances are measured by clocks. Light pulses form lines that connect null separated events. This is spacetime.
The lack of absolute space and absolute time means only that there is no preferred coordinate system on spacetime. There is still an underlying geometry where points and paths and distances can be identified and determined and all of those geometric quantities are independent of any choice of coordinates. The lack of absolute spacetime does not imply that the underlying geometry does not exist, but simply that any coordinates are equally valid.
A: It is perfectly reasonable to say that an event is a point in spacetime and that spacetime is a collection of events -- it is not "circular" as you claim in the comments. This is just the physics version of "a vector is an element of a vector space" and "a vector space is a set of vectors". You have axioms in math, and you have axioms in physics. The only difference is that in math, the objects are abstract, but in physics, they have a physical interpretation.
A: Think of the event as simply an occurrence, something that we all agree happens. If the idea of an event is confusing just take the definition to be what we normally use in everyday language. So an event is something that happens. Depending on where you are relative to the event when it occurs, you will have a certain time frame, unique to you. This happens because light has to travel the same speed at different distances to relay the event to the observer. So, in the unique timeframe of every "observer", the event is interpreted differently. The relationship between S and S prime, then, is not an equivalence, but a relationship that has to do with the time dilation between the two events. As for what spacetime "means", it's just a way of fusing our three measurable dimensions (x, y, z) with the fourth time dimension so we can calculate things like relationships between two time frames.
A: I would describe an event as something you can observe, at least in regard to Lorentz transformations. Of course it is also a point in spacetime, which means a point in Minkowski space, where the 3 dimensions of space and time are properties of one continuous entity.
Now what is the point of the Lorentz Transformation?
The starting point is the frame of reference of one observer with a specific location in spacetime. To calculate the view of the same event in another frame of reference, which means a different point in spacetime, you can do a Lorentz Transformation.
It allows for example the calculation of how the movement of an object ist percieved by observers with different locations and different velocities or roations in space.
Form the mathematical perspective you leave points in spacetime at a certain location, but you move the coordinate system. So if you moved the ccordinate system along the positive direction of the x-axis you will get lower positive values for x, but higher negative values for x. This can be done along every space axis and event time. Other possibilities are roation and reflection.
A: Spacetime is a collection of events, and, as such, an event is a point in spacetime. There is no contradiction whatsoever here.
Think of a classical $2D$ euclidean plane. This plane is defined to be a collection of points. You can take any point in the plane and assign it some $x$ and $y$ values depending on where you set the origin to be; if you move the origin to another position the values of the $x$ and $y$ coordinates of the point changes but the point itself doesn't.
An event is simply a point in spacetime, because specitime itself is defined to be a collection of points, called events. The fact that the same event, i.e. point in spacetime, can be described by arbitrary $x$ and $t$ values is a consequence of the principle of relativity, which states that no particular frame is unique.
The point can be chosen arbitrarly too and any two points are indistinguishable from each other, because of the assumption of homogeneity of spacetime.
Another, less formal but more physical, interpretation of event is that an event can be viewed as something that is happening, may happen or may have happened. In this case spacetime is defined to be the structure in which events happen.
A: I think I understand the dilemma here. Every idea in physics has two sides: the mathematical abstraction and the physical realization e.g. we have some intuitive notion of velocity in our heads through the physical world which can be accurately modeled and abstracted as some vector in Euclidean space (in classical mechanics at least). You want the latter and I think a better description of the former.
An event's mathematical abstraction is a point in Minkowski space-time (don't worry, I'll get to the physical realization in a moment). An important clarification here is we are not working in a vector space, per se. We are working in an affine space i.e. one with no origin (intuitively, this makes sense, space has no origin). This means there are not unique coordinates for any individual event (even in the same inertial frame)! The only concept that really makes physical sense is a vector between points not the vector of the points themselves. Lorentz transformations work only on space-time vectors and hence only between these differences of points i.e. $\Delta x, \Delta t$, etc. If you want to read more on this notion of affine spaces in the context of classical mechanics, I refer you to Arnold's beautiful book on mechanics. This leads us to change our train of thought from singular isolated events to pairs of events because only those have any relation to Lorentz transformations.
We know, from basic vector calculus, that vectors can be written in many different basis. The same applies to these Minkowski space-time vectors. Each inertial frame is simply a different basis. Sounds cool but doesn’t make quite that much sense yet, I know. It will become clear in a second when we talk about what an event really is physically. Lorentz transformations, are simply a basis transformation i.e. physically a frame change. Now to the moment we have all been waiting for. 
The physical realization. We have said above that we must think in pairs of events. Lorentz transformations help us see the vector between these two events (i.e. the spatial and temporal difference) in different frames. Now, we simply define an event as something that happens.
For example, a train entering and exiting a tunnel. This is an event. There is a frame-independent way of seeing if the train’s front is at the entrance and the train’s front is at the exit. Simply when they are aligned. Great! Now Lorentz transformations tell us that if we take the spatial and temporal difference between these two events in one frame, we can use Lorentz transformations to figure it out in a different inertial frame. Yay! 
Note that it is important that those events were frame independent. For length contraction, the two events are the measuring the position of the front of the stick and the back of the stick at the same time. However, in different frames these are actually different events because our measurement was frame-dependent i.e. we forced simultaneity.
I hope I answered your question explaining how we should think about events and their coordinates, what the Lorentz transformations mean physically and mathematically, and what events are physically and mathematically. Let me know if there is anything I missed!
A: There is a philosophical dimension to this question which i can't adress; i would like the following perspective, which starts from an observer. 
Special relativity is a theory which allows two observers to compare their results, which holds in the limit where neither quantum mechanics nor gravity is important, and they move relative to each other at constant velocity.
This has to be brought into formulas. We have to assume, that an observer $O$ can measure distances, and that they have a clock to measure time spans. In fact, usually one lets the observer measure distances with their clock, using the assumption that the velocity of light is constant. This can be formalized by saying that an observer carries a frame, which is their personal copy of $\mathbb{R}^4$. It's like a notebook they are taking with them. To stress that this is their personal copy (also called a chart), denote it by
$$(\mathbb{R}^4)_O \ .$$
Then the observer $O$ can cook up some way to associate numbers $\varphi(x)$ to points $x \in (\mathbb{R}^4)_O $. Since we have assumed that $O$ can measure distances, this coould be for example the location of their middle finger tip relative to their eyes.
If there is only one observer, special relativity is useless, so let's assume we have at least two observers $O_1,O_2$, and they have met sometimes in the past, and they agree on how to measure distances and time spans and have also agreed on one way of attaching numbers $\varphi$ to their copies of $\mathbb{R}^4$.
Then in particular, after this has happened, they can observe the relative position and velocity in their personal copy of $\mathbb{R}^4$. Let's say $O_1$ has measured that $O_2$ is at a distance $\Delta x$ and at a velocity $\Delta v$. Then $O_1$ can cook up a map
$$(\mathbb{R}^4)_{O_1} \ni x \mapsto f_{\Delta v, \Delta p}(x) \in (\mathbb{R}^4)_{O_2}$$
with which $O_1$ can relate points in $(\mathbb{R}^4)_{O_1}$ to points in $(\mathbb{R}^4)_{O_2}$.
This is in particular useful to relating their assignements of numbers $\varphi(x)$, since $O_1$ now knows that $O_2$ has assigned numbers
$$(\mathbb{R}^4)_{O_2} \ni x \mapsto \varphi \circ f_{\Delta v, \Delta p}^{-1}(x) \ .$$
Special relativity is a way of cooking up this function $f_{\Delta v, \Delta p}$.
Note that it is not clear what an observer is here, it is the fundamental concept. This is, as far as i can judge, what is called a neo-kantian approach. I can not define it, but i can point to examples.
A: Physically an event is something you have to be causally capable of "hearing" about or seeing. This means it must lie within an observer's past light cone. Otherwise if something was outside our  lightcone it would be meaningless to speak of (Einstein's elsewhere).
Geometrically this puts events on a higher standing that just any old points in spacetime.  Often in special relativity, a problem would be described regarding light signals and various observers moving around and statements like: 

"Observer A sees the event at time t"

What this is really saying is that the point where the event is, is just then becoming causally connected to observer A and therefore meaningful for them to speak of in terms of describing well...anything (maybe nothing) happening. 
In math speak a spacetime point first becomes an event to some observer, when a null-vector (lightlike movement) from the point first reaches them. If there's nothing worth mentioning it might be recorded as a null event.
Everything within the observable universe is an event. Note that this definition automatically excludes the interior of black holes as events, which must be why they call it an "event" horizon.
A: Many of the answers already here raised some good points [no pun intended].
I'll offer my take in trying to answer the questions in your EDIT.
(Since the question is vague, 
my comments [like those of others] are trying to anticipate where your concerns may lie.)
I'll respond with a numbered list so that they are easier to reference.


*

*When issues concerning "spacetime" and "spacetime diagrams" are raised, 
it is useful (for comparison and contrast) to see how they are addressed in


*

*ordinary diagrams of space (using Euclidean geometry)

*position-vs-time diagrams from PHY 101 (which has an unappreciated Galilean spacetime geometry)



(I think it's fair to say that Minkowski's "Space-Time" and "Space-Time Diagrams" are based on these. If one is okay with what goes on in the diagrams above, then one should be okay with what happens on a spacetime diagram. Specifically, Are you okay with events on a Position-vs-Time graph? If not, then the specific issue has to be articulated.
NOTE: We create "space"-"time" diagrams because we are interested in the relations (temporal, spatial, and especially causal ones) among [event]points in spacetime. 

If we are interested in something else (like energy and momentum), we use a different diagram (like the energy-momentum diagram). What you hope is that your diagram will help you answer your questions.
)




*"Events" in special relativity and in galilean relativity are like 
"Points [as a pencil dot]" in an ordinary Euclidean diagram...
in that they mark a mathematical idealization of a point with no extent in any "direction" (possibly motivated as the limit of a sequence of smaller markings).



*

*Note that one can have different types of marks for the same point. Just as we can have a dot by pencil-tip or chalk-mark or the intersection of two lines (or all three) in a Euclidean diagram,... in relativity [special or Galilean] a "point in a spacetime diagram" could be marked by a firecracker or a snap of a finger or a lightning flash on a track (or all three) (again, each representative of a sequence of quicker and smaller ones).

*"event" are "point" are often used synonymously.
 For convenience [although it may be sloppy], 
we often refer to a convenient mark instead of the point it represents. 

*We emphasize "[events] points in spacetime" are things that potentially could be marked by something (say) a finger-snap... it doesn't matter what it is... it could be real or hypothetical. (In Euclidean, we often refer to "arbitrary points" that are not (say) the intersection of two given lines... but we can construct two lines that intersect there.) 

*Spacetime is not interested in whether a particular kind of mark [i.e. if a certain type of event] can or cannot be made. A point in spacetime can be marked by something--real or hypothetical. If it can't, it shouldn't be part of the spacetime!  

*Everyday use of "event" (like a basketball game or a clearance sale) are generally not appropriate. Consider the limit of smaller and shorter-lasting versions.


*In Euclidean 2-D space diagrams, we often say we label points with (x,y) coordinates 
because we can "lay down two axes" (typically perpendicular to each other)


*

*so that we can [via some procedure] assign a unique pair of numbers to LABEL each point

*and we do this "laying down of axes"
with the freedom that there are no absolute-x and absolute-y directions.

*using other axes will results a different labeling of the same points, and transformations that preserve the Euclidean structure (rotations, translations, and reflections) will show how to convert the labelings from one set of axes to another

(NOTE: One could imagine a situation [by convention or by history] where the x-axis must be measured in meters by reading a ruler mark and the y-axis must be measured in miles by the length of a line of string cast from the meterstick.)


*(elaborating on 3.)
In a position-vs-time diagram [in Galilean and Special Relativity], we use two kinds of axes--one using a wristwatch and the other either a meterstick or a line-of-string or "something similar" (whichever is convenient, since it may be impractical to lay out a ruler to the moon or too-complicated to analyze such a setup to be sure that it is properly making the intended measurement).



*

*We use two types of axes because it turns out that there are two different types of lines (timelike and spacelike).
Observer worldlines are only along timelike curves, and with a common orientation [the future direction]. This is different from what happens with spacelike curves.

*Note that there are multiple time-axes, each corresponding to the tangent-vector of an observer's worldline.

*Note (generally) that, at each event, there are multiple x-axes, each corresponding to a vector "perpendicular" to an observer's worldline.


*(elaborating on 4.)
It turns out that in Galilean relativity (extrapolating results of "experiments done with slow objects" to hold when much faster objects are involved), all wristwatches (within the limits of their resolution) seem to give the same measurement. So, it might be more convenient to maintain a universal clock that each observer uses [and doesn't question].


We [commonly] refer to events in a [Galilean] position-vs-time by labeling with $(t,x)$-- a time-reading and a 1-D location.


*

*There is absolute time here... and it seems we declare the wristwatch "at rest" in our frame to determine the "time axis" and be the universal clock.

*There is no absolute space here in the sense that the "spatial distance between two general events [i.e. occurring at different times]" depends on the frame of reference. (Nothing weird here: In Euclidean geometry, the y-difference between two points depends on the orientation of the reference axes.)

Note: "absolute time" (the "same time measurement") actually implies that all Galilean x-axes are actually parallel to each other. So we declare the x-axis of the observer at rest to be the "spatial x-axis" direction. 

[Perpendicularity is defined by the tangent-line to the "circle" (the curve of equidistant points from a given point). The radius and tangent line are mutually perpendicular.]



Here is a good time to pause.
Does the OP have any issue with item 5 (built from items 2, 3, and 4)?


*In analogy to Euclidean 2-D space diagrams (with no absolute-x and -y) in item 3, and with issues raised in items 4 and 5,
in Minkowski (1+1)-D spacetime diagrams,
 
we often say we label 
[event]points with $(t,x)$ coordinates 
because we can "lay down two axes"


*

*so that we can [via some procedure] assign a unique pair of numbers to each [event]point

*and we do this "laying down of axes"
with the freedom that there are no absolute-t and absolute-x directions.

*using the wristwatch on the observer's worldline to provide the first axis,
and the "perpendicular" direction to this worldline as the second axis.

*using other axes (e.g. different wristwatch worldline and its corresponding perpendicular direction) will result in a different labeling of the same [event]points, and transformations that preserve the Minkowski structure (boosts-and-rotations, translations, and reflections) will show how to convert the labelings from one set of axes to another


*Elaborations:


*

*We emphasize:
we use a $(x,y)$ labeling even though Euclidean space has no absolute directions
and so,
we use a $(t,x)$ labeling even though Minkowski spacetime has no absolute-time or space

*the coordinates $(t,x)$ are just the unique labelings using one set of axes... and you could get to other labelings from other axes with the suitable transformation equations... or you could collect all labelings and somehow group them accordingly [into equivalence classes] and deduce the transformations.

*one could use light-cone coordinates $(u,v)$ which could be interpreted as labeling using two time-readings from an observer's wristwatch in radar-measurement.
(Technically, we only get magnitudes for spatial coordinates... we need a sign from knowing the direction of the light rays in radar experiment.)

*one could use any appropriate method that generates a unique labeling of points in spacetime... just give a operational definition that describes your measurement.
From Synge's Special Relativity [p. 7],


In this spirit, let us see how the space-time coordinates $(x^l, x^2 , x^3 , x^4)$ 
  may be assigned to an event. Suppose that the event is the explosion 
  of a rocket in mid-air. Let there be four observers, flying about in 
  aeroplanes, not on any particular courses, but turning and diving and 
  climbing in an arbitrary way. Let each observer carry a clock, not 
  necessarily an accurate clock, but perhaps an old battered clock -- the 
  one essential is that it keeps going.
 
  Each observer notes the reading of his clock when he hears the 
  explosion of the rocket. Let these four readings be denoted by 
  $(x^l, x^2 , x^3 , x^4)$; these four numbers may be taken as the coordinates 
  of the event. It is clear that if there are several events 
  (several explosions), they will in general give distinct tetrads of numbers, and in 
  fact there is a one-to-one correspondence between possible events and 
  possible tetrads of coordinates, with perhaps the exception of certain 
  critical events. 
  

  ...The essential point is 
  that it is possible to give operational procedures for the assignment of 
  coordinates $(x^l, x^2 , x^3 , x^4)$ to events in space-time, and that there is an 
  infinite variety in the ways in which this can be done. If two different 
  procedures are used, the first giving coordinates $(x^l, x^2 , x^3 , x^4)$ and the 
  second giving coordinates $(y^l, y^2 , y^3 , y^4)$ to the same event, 
  then there 
  will exist formulae of transformation of coordinates... 

added


*Given two spacetime diagrams, how does one identify the same events?
With a labeling of events on the first diagram (say, events A,B,C, with corresponding pairs $(t_A, x_A)$, etc...), see how those pairs $(t_A, x_A)$ transform... then give the A-label to the transformed event.

The transformed diagram also carries geometrical information that is preserved by the transformation. For example, the square-interval between A and B in the original diagram is equal to the square-interval between the transformed events.

*A spacetime diagram can help you analyze the temporal, spatial, and causal relations between events (marks) of interest to you. 


For example, if you had two separated detectors (at rest for simplicity) that clicked: detector-A at t=2s and detector-B at t=5s. You could use a spacetime diagram to analyze where the light-source may have been.


From all of the details of the setup, you use only the position and time information... and then you work mathematically with the points on the diagram to find the point that would correspond to the source-emission-event. The diagram, the geometry, etc... just care about the points and their coordinates---without caring about what made the marks.

(You could invent a totally different story with different objects that could lead to the same points of interest. This whole discussion is similar to saying that once we formulate an equation modeled on the physics, we just solve it [without immediate regard to how we got that equation]. Then, use the mathematical results to interpret physically. Later we might ask about how accurate our modeling of the problem was... but that's a different problem.)
A: First of all, I would not recommend to follow the definition of an event as a "point in spacetime". A point is a point, and an event is an event, and at points where nothing is happening you can hardly speak of an event. For this reason, I will talk about "particle events" which are meeting points of the worldlines of particles. An inertial frame will observe a particle event at (x,t) only if a light beam has been sent to this point, and if the light beam has hit some particle.
Particle events play an essential role within special relativity because they are invariant and observer- independent. That does not mean that all observers are measuring the particle event at the same coordinates, but the invariance is concerning the mere fact that there is a particle event.
Particle events are invariant in the same way as proper time, so, if you distill the invariant phenomena of spacetime, spacetime is represented by all events which are linked among each other by some proper time. However, every observer will observe events at a different place, and he will not observe the proper time but space and time intervals.
