Special Relativity Could someone explain to me how special relativity works?
I know there are thousands of sources and databases of knowledge out there, but I find it difficult to understand, even after reading up on those sources.
(Note: if you're an admin to close my question down, would you please be so nice as to point out something to help me with this question?)
 A: I think I could give some intuitive look on SR. It is not very hard to understand the basic overview of SR. There are only two postulates and not more that that..! But, there are many sites which provide a bit wrong infos. And, Whooo - found it... Your question is just a duplicate.
First postulate: "The laws of physics hold good (are the same) in all inertial frames of reference". 
First of all, SR declares that all motions are relative. Mass, length, space & time, etc. are not independent but they are all inter-dependent according to Einstein's view which discarded absolute space and for now, it is treated obsolete.
Say for example (the most basic one) - You and your friend are traveling in parallel but exactly the opposite direction at speeds $v_1$ and $v_2$. Relative to you (you are a rest frame now), your friend would be past you at $v_1+v_2$. Both of you would experience some effects (mentioned below) and would also measure different distances, time, etc. The physical laws would be the same because you guys are in an uniform motion. If both were racing each other, one would measure the other's velocity as $v_1$ ~ $v_2$. And, another thing I'd like to note - If both travel near $c$, you'd have to take Lorentz factor $\gamma$ into account. So, you'd have $\Delta l,\Delta t, \Delta m$ and even relativistic acceleration. But, these are noticeable only to the worst cases (like above $0.5 c$).
Second postulate: "The speed of light ($c$) is the same in all inertial frames of reference".
Wherever you both go, you guys will measure the speed of light to be the same value $c$ because you guys are still in inertial frame. This postulate is perhaps given a greater priority because it specifically says that information could not be transferred at velocities above $c$.

Thus, SR concluded some new facts like slowing time, shorting length, apparent mass, Could Faster than light be possible?, got stuck by a twin paradox, the possibilities for faster than light, etc., etc... A great thing to note is - all these effects could be experienced by you only if you travel comparatively near $c$. Oh... And the most important ones - Mass-energy equivalence, Space-time and Lorentz transformation.

For more info, please refer the duplicate one. A best reference would be a Simulation. Once you have some basic understanding on SR, please see Real Time Relativity. It is totally amazing. Of course, I found it (some time ago) in Lubos' blog.
Note: Gravity also affects objects. Actually, the effect of gravity on objects were generalized by Einstein to expand SR to GR (took some years though) 
A: I agree with Crazy Buddy that Lay explanation of the special theory of relativity? is a good approach to SR, but my own preference is to view it a bit differently. You probably heard it said that general relativity is a geometrical theory. Well special relativity is a geometrical theory as well.
If you take two points in Euclidean space ($x_1$, $y_1$, $z_1$) and ($x_2$, $y_2$, $z_2$) and denote $x_2 - x_1$ by $dx$, $y_2 - y_1$ by $dy$ and $z_2 - z_1$ by $dz$, then the distance between the points, $ds$, is simply given by Pythagorus' theorem:
$$ ds^2 = dx^2 + dy^2 + dz^2 $$
and the distance $ds$ is an invarient. We can rotate or translate our co-ordinates, or travel at any speed we like, and we'll still calculate the same value for $ds$. This is all pretty obvious, for example $ds$ might be the length of a metal rod (with the two points at its ends) and in Euclidean space the length of the rod isn't going to change.
To move to special relativity all we have to do is change the equation we use to calculate the distance between the spacetime points ($t_1$, $x_1$, $y_1$, $z_1$) and ($t_2$, $x_2$, $y_2$, $z_2$) to be:
$$ ds^2 = dt^2 - dx^2 - dy^2 - dz^2 $$
and insist that the line interval $ds$ is an invarient i.e. all observers will calculate the same value for $ds$ no matter how fast they're moving. This simple principle then gives all the weird effects we see in SR.
I call this a geometrical approach because it's the SR equivalent of Pythagorus' theorem. It's just a prescription for calculating the distance between two points.
Whether this is helpful, or maybe just even more confusing I don't know, but you can see how this gives results like a finite speed of light by looking at my answer to What is the relationship between the speed of light and virtual particle production
A: Study of special relativity can be divided into two parts -- 
$1$. Mathematical notions - You need to know about Minkowski space and Lorentz group.  Minkowski space is the space $R^{1,3}=\{(t,x,y,z)|t,x,y,z \in R\}$ of 4-vectors with a scalar product defined as $(t_1,x_1,y_1,z_1).(t_2,x_2,y_2,z_2)=t_1t_2-x_1x_2-y_1y_2-z_1z_2$. Lorentz group is group of those matrices $\Lambda$ which satisfy $(\Lambda X).(\Lambda Y)=X.Y$ for all 4-vectors $X$ and $Y$. 
These mathematical objects can be understood in straightforward analogy with Euclidean 3 space $R^3$ with usual dot product of vectors and the group of 3 by 3 orthogonal matrices which preserve this dot product. Lorentz group which acts on a larger space is of course larger than the usual orthogonal group. Other than matrices which only rotate $x,y,z$ among themselves keeping $t$ fixed, Lorentz group also contains matrices which mix $t$ with other coordinates; these latter Lorentz transformations are called boosts.
$2$. Physical notions
i) Space time - Space time is collection of events. As a set it is same as Minkowski space (without any specific choice of coordinates)
ii) Inertial frame (or inertial observer) - Physically the notion of inertial frame is explained well on Wikipedia. Mathematically an inertial frame can be thought of as a coordinate system on $R^{1,3}$ which is related to the usual coordinate system (which itself is assumed to be inertial) via a Lorentz transformation. If $O$ is an inertial frame then (by the mathematical definition) it follows that a frame $O'$ obtained from $O$ by a rotation of spatial coordinates and/or by a boost will again be an inertial frame. Those inertial frames which are related to each other via a boost transformation are physically interpreted as moving relative to each other.
iii) Event - An event is a point of space time, that is an abstract point of $R^{1,3}$. The coordinates assigned to an event depend upon the observer who observes it (i.e. the inertial frame in which it is observed). 
After understanding these notions one formulates the principle of special relativity as -
"The laws of physics are same in all inertial frames". More precisely physical quantities and physics equations should be Lorentz covariant i.e. should be written in terms of scalars, 4-vectors, and tensors. Speed of light is assumed to be a scalar (i.e. a constant in all inertial frames).
Special relativity is special in the sense that here we only work with inertial observers. In other words we use only those coordinate systems on our space time which are related to each other via linear transformations. When this restriction is lifted and hence all types of coordinate changes are allowed then we get general relativity. There the space time is a manifold with a metric. An event is an abstract point of manifold. And notion of inertial frame is generalized to that of a local coordinate patch.
