First, a little bit about Einstein’s Special Theory of Relativity. It started with two postulates: 1) that the laws of physics are observed to be the same as long as you aren’t accelerating (i.e., you’re moving at a constant velocity) and 2) that the speed of light c is observed to be the same for all observers.
In special relativity, one is always comparing what happens in two (or more) “reference frames,” moving at some constant relative velocity to one another. Shortly after he published the special theory in 1905, Einstein published a second paper about a particular result. Basically, he imagined what would happen if you had an object with a certain mass emitting light—then you observe that object from a frame of reference in which it’s at rest and a frame of reference in which it’s moving with some constant velocity. He found that the energy carried by the emitted light is different in those two cases. But the total energy has to be conserved (i.e., it has to stay the same in both frames of reference). Where does the additional energy come from? From the kinetic energy of the object.
Kinetic energy is simply energy of motion: it's proportional to the mass of the object and the square of its velocity. But we know how fast the object is going in both reference frames because that’s built into the problem. So the energy must result in a change in the mass of the object! Some tiny mass m is given up to provide the additional energy; it turns out that it does so proportional to the speed of light squared.
Thus, you end up with the formula E=mc^2, which results from comparing the same physics problem from two different perspectives in special relativity. It turns out that this has great implications for quantum mechanics (the study of nuclear energy, for example), but it originated from relativity.
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