E=mc^2 may be the most famous physics equation in history. Why this is, though, is misunderstood, both by the public at large and even by many physics students (at least ones I've talked to about this).
So, the Public Understanding: Einstein was a super-genius, and he invented E=mc^2. This has something to do with energy. Einstein used this to invent the atomic bomb and win World War II.
Why this is Wrong: Well, Einstein was actually a super-genius. I kind of have a crush on him, to be honest. And he did derive (an important point we'll come to later) E=mc^2. He did not, though, have much to do with inventing the atomic bomb. What's more, E=mc^2 didn't lead straight to the bomb in the sense that most people think it did.
So let's take a step back. The equation we're talking about says that Energy (E) is equal to (=) mass (m) times the speed of light (c) squared (^2). This tells us that (a) mass can be converted to energy, and energy can be converted to mass, and (b) a little bit of mass converts to an enormous amount of energy, since c^2 is a very big number. Now, it's certainly true that the mass of the final nuclei involved in a nuclear bomb is less than the mass of the initial nuclei, and that this change in mass is proportional to the energy released. But that's true of all processes. When I burn gas in my car, the final products are ever so slightly lighter than the initial ones. But I don't credit E=mc^2 with making my car run. So what's up?
The reason we associate E=mc^2 with nuclear (ie, a-bomb) processes and not chemical (ie, gas-burning) ones is basically a matter of technical convenience. When I burn gas, it's easy to measure the energy that came out, but hard to measure the change in mass, because it's incredibly tiny. When I split or collide nuclei, it's hard to measure the energy that comes out, partly because there's so much of it and partly because a bunch of the energy gets carried off by neutrinos, which we can't capture very well. But it's (relatively) easy to measure the mass of the initial and final nuclei, so that's what we do. E=mc^2 is always true, it's just sometimes convenient to use, and other times not.
In any case, most of the effort that went into building the atomic bomb was on rather practical questions like, "How do we separate out the uranium we want from the uranium we don't want?" and "How can we use precision explosives to bring that uranium together in just the right way?" These questions had really nothing to do with E=mc^2.
Now the Common Physics Student Understanding: Einstein was a super-genius, and he derived E=mc^2. This tells us that mass and energy are equivalent, two aspects of the same thing. This changed our view of reality.
Why this is Wrong: Well, it's really not. What it is, though, is incomplete. So to complete it, we have,
Why E=mc^2 is Cool and Important: To understand this, we need to take a look at where the equation came from. Where it came from was two papers Einstein published in 1905 on electricity and magnetism. Einstein starts off this little duology by noting that, at the time, the laws of electricity and magnetism were inconsistent with the laws of motion in a peculiar way. The example he used requires a bit of background, so I'm going to pick a simpler, but equivalent one.
You probably learned at some point that electric current can make magnetic fields--this is how we get electromagnets. In fact, any current, and any electric charge that's moving, creates magnetic fields. This, though, creates a problem. Say I rub a balloon on my head to put some charge on it, then put my charged balloon out in space, a long way away from anything. Now, if the charged balloon is moving, it creates a magnetic field; if it's not, it doesn't. But, how do we know in space what is moving and what is standing still? If one person (normally called Alice) is floating next to the balloon, and another person (normally called Quvenzhané) shoots past, they would disagree on who is moving and who is standing still, and hence they would disagree on whether or not the balloon was producing a magnetic field. But the magnetic field can't both be there and not be there, so we have a problem.
Einstein noted this inconsistency, and found a way to write physics laws in a way that didn't create these disagreements. It was a bit of a weird way, with time slowing down as you sped up, and lengths changing and whatnot, but it worked. And, almost as an aside, it produced the expression E=mc^2.
The details of how that all works aren't really important here. What is important is this: the laws of Electricity and Magnetism (EM), which you can work out with some styrofoam balls and plastic in a high school classroom, imply that every object in the universe has an intrinsic energy that only depends on its mass. Not its internal structure, of what it's made of, just its mass. So E=mc^2, which isn't really about EM, and applies to things that aren't charged or magnetic, and plays a large role in gravity, is embedded in the structure of electricity and magnetism. This should blow your mind. The laws of how electricity works also tell you that everything has an intrinsic energy proportional only to its mass. This is one of the best pieces of evidence so far that there is, in fact, a consistent mathematical structure underlying the universe. That Einstein figured out this implication pretty much cemented his genius status, even if he didn't single-handedly win World War II.
And THAT is why E=mc^2 is cool.