Significance (if any) of coordinates in GR I've studied General relativity for some time now and Read a lot about it, but still I'm convinced I am missing (at least one of) the main point. Maybe the simplest way to put my question is: what coordinates would I refer to as the coordinates of a freely falling observer? Does this question even make sense in this context? Are the coordinates that define an observer or is it something else? To make an example take the Schwarzschild Solution in Schwarzschild coordinates ($t, r, \theta, \varphi$): the question would be who is using those coordinates? Are them referred to a particolar observer? In general if I have a set of coordinates in a spacetime with assigned metric, how can I determine if the coordinates I am given are the one of a inertial (freely falling) observer? I know this may be a ill-posed question... 
 A: There is no physical significance to coordinates in GR: coordinates are labels we use to describe where and when events happen, but there are no coordinates in the world.  Coordinates are part of the tools we use to describe the world, not part of the world.
So you can use any coordinates, so long as they are 'good' to describe any system.  Here 'good' means that they must actually provide a 1-1 mapping between spacetime and $\mathbb{R}^4$ and that mapping must be suitably differentiable.
That said, there are choices of coordinates which make it much easier to work with a particular system than others, and in practice this means that you want to work pretty hard to choose the right set of coordinates: ones which respect the symmetries of the system in some way, or ones whose timelike axis correspond to the worldline of some observer you are interested in, say.  Not doing that can make it absurdly hard to work with a system.
So in practice, coordinate choice is very important -- just as it is in Newtonian physics! -- but it's important to understand that coordinates are just a bookkeeping mechanism we use to describe the system, they're not something that is part of the system itself.
A: The way we can determine whether a coordinate system is one of a freely falling observer (the same as being locally inertial) is by calculating the Christoffel symbols. If all Christoffel symbols are 0 at point $p$, then the coordinates define a locally inertial frame around $p$ (note that in general the derivatives of the Christoffel symbols are not zero). A reference on this: http://www.theoretical-physics.net/dev/relativity/relativity.html#inertial-frames.
In general, coordinate systems in GR have no correspondence to what an observer would "see", they are simply mathematical tools used to describe a set of points with a set of ordered pairs. Sometimes coordinate systems, such as the Schwarzschild coordinates, have obvious physical correspondence in part of the domain (far outside of the black whole $(t,r,\theta,\phi)$ are simply spherical coordinates with a time coordinate), but are more abstract in different regions (inside the event horizon $t$ becomes a spacelike coordinate, so could not possibly be the same as the proper time for some observer). 
A: So in general relativity we're doing physics with a branch of mathematics called differential geometry. One of the things you have in your physics is called a "manifold." And when you want to interact with it (or any mathematical object), you need to ask the mathematicians for the axioms, the postulates that the object satisfies simply in defining what we're talking about, and maybe also lemmas and theorems built from those axioms: and then you build your physical queries out of those axioms, lemmas, and theorems. In doing so you build a "model." The "theory" of general relativity is all about applying the mathematics of differential geometry to construct models of gravitational phenomena.
So the mathematicians tell us that the basic unit of differential geometry is the manifold, and the definition of the manifold requires certain axioms: that the manifold has a space of otherwise-arbitrary "points" $\mathcal M$ and a set of 'smooth' scalar fields  $\mathcal S$ which all have type $\mathcal M\to\mathbb R,$ they all map these points to real numbers. And you get more axioms -- it turns out that these scalar fields are called 'smooth' because they are closed under certain axioms. And theorems -- it turns out that the scalar fields define a 'topology' -- a notion of what points are "in the neighborhood" of another point. So the mathematicians are telling you all of these axioms and theorems you might use, and finally, they also tell you the coordinate axiom, which goes something like this:

The manifold is $D$-dimensional for some natural number $D$. This means that for every point in $\mathcal M$ there is a neighborhood about that point, and scalar fields $c_1,c_2, \dots c_D$ that we call coordinate fields in that neighborhood, such that:
  
  
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*If two points in this neighborhood are different, at least one of the coordinate fields is different when evaluated at those two points,
  
*On this neighborhood any smooth scalar field $s(p)$ can be re-understood as a smooth function $\sigma \in C^\infty(\mathbb R^D,\mathbb R)$ applied pointwise over the coordinate fields, so that  $$s(p) = \sigma\big(c_1(p),~c_2(p),~\dots c_D(p)\big).$$

These sorts of mathematical things need to be understood with some examples.
So for example if we have a 2-sphere -- the surface of the 3D ball -- they would say that that's a manifold. There are scalar fields $x(p), y(p), z(p)$ that give us the Cartesian coordinates of the points $p$ on the sphere. It is a 2-dimensional manifold because, take any point: let's say it's in the upper-hemisphere. Then a valid neighborhood is the upper hemisphere, the coordinate fields that work there are $x, y,$  and every smooth scalar field on the sphere can be understood on the upper hemisphere via these two fields. "What about $z$?" you might ask -- well $x^2 + y^2 + z^2 = 1$ means that we can understand $z$ as a smooth scalar field on the upper hemisphere, $z = \sqrt{1 - x^2 - y^2}.$ And on the lower hemisphere $x, y$ also suffice. The only place these two fields don't suffice is on the equator itself, but there we can either choose $x, z$ or $y, z$ plus an appropriate hemisphere oriented the other way. At any given place there is a neighborhood which these can be used for. But if you use these coordinates you find that there is an equator where they form a "coordinate singularity" -- they cease to be helpful at the edges. They also don't really correspond to anything that you really need in navigating the space, unless you happen to be at the North or South poles: if you think about the map you're creating here, it corresponds to drawing a plane through the equator and just "dropping" every point onto the map vertically, so that the sphere is mapped within the unit circle. 
"Why are you going through all of this?" you might ask. "I know a better way, use the spherical coordinates $\varphi, \theta$." Well not so fast, the "latitude" angle that sweeps down from the pole is fine, it's equivalent to the $z$-field -- but the angle that goes around the sphere is not fine because it's not smooth: adding it to that list of smooth scalar fields messes up the topology something fierce because points near the $2\pi\to0$ crossover become disconnected and you end up only having a sort of manifold that looks like a rectangle or maybe the profile of a football, rather than a sphere. 
But you are not far off: there are many other ways to do coordinates of the sphere. One nice one is, if you deform the above "map" a little bit so that you do not simply drop the points straight down, you will find that you can continue the map past this horizontal boundary. The easiest way is to "drop" them instead along a straight line towards the South Pole, forming $x/(1+z)$ and $y/(1+z)$. Then you do find that every point except for the South Pole itself can use the same coordinate fields. (In some sense this is what is wrong with Schwarzschild coordinates too; they are based on a system of nested spheres and those spheres happen to be paradoxically behaved at the event horizon.)
So in summary: in GR our "coordinates" are (1) not necessarily valid over the entire space, and (2) they do not directly measure any particular distance. They instead label the points in a way such that we can then use the metric tensor to measure distances. 
As an example of (1), famously if you look at a Schwarzschild black-hole using some coordinates that are natural at long distances, you find another "coordinate singularity as you approach the black hole, before you reach the essential singularity at the very center. Many people misinterpreted this singularity for a long time, as saying "it will take you an infinite time to fall into a black hole," but we now understand that it is a coordinate singularity and that there are other coordinates, like Lemaître coordinates, that continuously pass through it. While it might appear to take forever from a long distance, it appears to happen in finite time when you're falling in!
