Where E is energy, m is mass, and c is the speed of light. It's a very simple-looking equation with only three parameters, but what does it mean? Well, it means that anything with mass – you, your cat, your house, the Earth – has latent energy stored in it, and the amount of mass determines that latent energy. For an object at rest, this correlates to the rest mass of the object. If an object is moving really fast (near the speed of light) its kinetic energy causes it to actually get heavier, since the object can never actually reach the speed of light (only objects with no rest mass move at the speed of light).
So, if we have an object sitting and doing nothing, and it suddenly glows for a split second, then stops, where did the light come from? Well, light has energy, as we know, so we could calculate the energy of the light that escapes our object. If the light emanates in all directions, then the net kinetic energy of the object is unchanged. But conservation of energy says that energy can neither be created nor destroyed! Have we violated the laws of physics with our weird glowing object? Well, no, because if you were somehow able to weigh the object pre- and post-glow, you would find that the mass of this object is actually slightly less after the light is given off.
But wait! Doesn't conservation of mass say that matter can neither be created nor destroyed? Well, yes, it does say that. So the only way for this to make sense is if the mass is converted into the energy that was emitted. We know that energy can be converted into different forms (electric, mechanical, thermal, etc.), so this must mean that mass is another form of energy that can be converted to and from! Pretty neat, huh?
Minutephysics has a cool video on this with a bit more technicality and pretty pictures of radioactive cats, but this is my text-based explanation simplified.
Another thing that may cross your mind is that this looks very similar to Newton's second law:
So, does Newton's second law equate force with acceleration? Well, no, because in the mass-energy equation, the constant of proportionality, c2, is a universal constant; it is the same for any and all objects in the universe. The mass of an object, however, varies from object to object, and is thus not a fundamental, universal constant, so while these equations are similar and relate two seemingly different entities, they do not conceptually perform the same task.