Part 0: More historical background
Part 1: The problem
Part 2: Space-Time
One of the results of SR is that velocities do not add. If one rocket is going to my right at 3/4 the speed of light, and another is going to my left at 3/4 the speed of light, these two speeds to not sum to 3/2 the speed of light. Instead, each rocket observes the other to be going at 24/25 the speed of light. The equation for the relativistic "sum" of two speeds is as follows:
(v1 + v2)/(1 + v1v2/c2)
...where v1 and v2 are the two speeds, and c is the speed of light
The derivation of this equation unfortunately relies on math that I skipped, so you'll have to take my word for it. It basically arises from the equations that fully describe lorentz contraction and time dilation. One thing to note is that if either v1 or v2 is equal to c, then the relativistic sum is equal to c. That means that no matter how fast an observer is moving, light will still travel at speed c. Einstein originally set out to explain why light is always observed to move at speed c, and now we've done it.
Also note that if v1 and v2 are both less than c, their relativistic sum will be less than c. So no matter how much you try, you can never push an object above speed c. In fact, what happens, is that the closer its speed is to c, the harder and harder it becomes to increase its speed. That means that Newton's second law (Force equals mass times acceleration) is wrong. At high speeds, it takes greater force for a smaller amount of acceleration. In some sense, the mass of the object increases when it speeds up. Specifically, the "relativistic mass" of the object increases. Usually when physicists talk about mass, by default they refer to "rest mass" (the mass of an object when it isn't moving) rather than the relativistic mass. The rest mass stays constant during acceleration.
Whenever you apply force to a moving object, you are transferring energy. You are also changing the object's relativistic mass. With a bit of calculus, we can conclude that energy is relativistic mass multiplied by a factor of c2. That's where we get the equation E=mc2.
Technically, the famous equation is incorrect. The correct form is E=ɣmc2, since these days, relativistic mass is written as ɣm. The letter m represents the rest mass of the object, though in Einstein's time, it was used to represent the relativistic mass, thus the source of this confusion. The number ɣ (represented by the letter gamma) is called the Lorentz factor, and it turns up in the equations for Lorentz contraction and time dilation. For the interested reader, ɣ=1/sqrt(1-v2/c2).
For a non-moving object, ɣ=1, but when the object's speed approaches light, ɣ approaches infinity. That means that E=mc2 is only true for an object that is not moving. In order to push an object to the speed of light, it would require an infinite amount of energy, and the object would gain an infinite amount of mass. Along with time-travel, this is why the speed of light is the ultimate speed limit. (However, this does not disprove the existence of tachyons, which, in theory, already start out faster than light, so that there is no need to push them through the light-speed barrier.)
I don't think most people realize the implications of this equation, so I want to expound upon them. Most people have heard that this is how atom bombs work. When an atom bomb explodes, the nuclei of Uranium atoms split apart. The way they split is such that the total mass decreases. Because c2 is such a large number, a small difference in mass results in a huge amount of energy. The same basic idea is behind hydrogen bombs, and nuclear power plants.
What most people don't realize is that mass-energy equivalence is true of all energy. If you use energy to heat up an object, it becomes a tiny bit heavier. This should not be surprising, since the temperature of an object tells you how fast the individual particles are moving, and I've already said that objects moving at higher speeds become more massive. But more surprising is that if you compress a spring, giving it potential energy, it becomes heavier. If you snap your fingers, you become slightly lighter, since a small amount of energy is released through the snapping sound.
Mind you, none of these mass differences come in large enough quantities that we would ever notice them, and they would be completely swamped by other effects. But mass-energy equivalence also gives protons and neutrons the vast majority of their mass. Without it, atoms would be too light, the force of gravity would be less effective, and we probably wouldn't exist. It is by the grace of Special Relativity (along with a ton of other cool physics) that we're all here.
This concludes my series on Special Relativity, though I may still talk about it, and go off on other tangents. If you've got any questions, ask, and I could write more to explain the answers.