I think it would be nice, for the benefit of all who are reading this and are still immensely confused, to explain the units that would be used with E=mc^2, in spite of all the purists who keep yelling about how it’s an equation that needs no units and that laypeople should just shut up. (After all, the worst answer to a scientific question, aside form a wrong one, is one that does not answer it.)
An equation such as E=mc^2 can naturally take any units it wants. In the metric system, however, the most likely units to be used are kilograms for mass and meters/second for speed of light. By multiplying these units, one would assume that E was expressed in kilograms-meters^2/second^2.
Such a unit is called the Joule.
Somebody asked what an hour squared was. Of course, hours squared are not real things (after all, I hope we only have only one time dimension to deal with, not two). Hours and seconds are squared not for a geometric or a physical reason, but for a mathematical one. Going back to the Joule, let’s try to understand what this odd unit of energy means.
According to the classical definition, energy is the ability to do work. Work is most simply understood as pushing things, and so it’s logical to measure energy in terms of how much energy you use up pushing (or, in more precise language, accelerating) something.
So you start off with a mass, measured in kilograms. Acceleration is measured by how much speed is increased per every second that passes. A car that goes from 0 to 60 in 6 seconds accelerates by 10 mph per every second. Speed itself is measured in meters traveled per seconds spent traveling, ie meters per second. An important thing to remember is that when a quantity is per another quantity (as in “there are a number of socks per a number of people”), it is understood as division. So going back to physics: speed is expressed in the unit m/s, and acceleration in a unit Speed/s. Via substitution, we get that the acceleration must be expressed in m/s^2. Seconds squared are not squares made of time, but are simply the algebraic end result. Also remember that m in m/s and m in mc^2 mean different things. In one, ‘m’ is an abbreviation for ‘meters’, in the other it is an abbreviation for ‘mass’.
So now we have the mass (in kg), and how much it is accelerated by, or how hard it is pushed, (in m/s^2). Now all we need is to specify how much this mass was pushed. We can either say it was accelerate for a certain amount of time, or for a certain distance. “For” has the mathematical meaning of multiplication (as in “a number of apples sold for a number of dollars each”, the two numbers must be multiplied). So the result is either m*s/s^2 or m^2/s^2. The first gives you an urge to cancel out. However, you cannot do this since you are dealing with units and not merely numbers. This makes the first possibility counter-intuitive and, moreover, not what we are looking for. Therefore, we choose to use acceleration for a certain distance.
So there you have it. kilogram-meter^2/second^2 means a unit of energy, a Joule, corresponding to the energy that is required to accelerate one kilogram of matter by an acceleration of one meter per second per second, for a distance of one meters.
What’s interesting is that E=massc^2 is very similar to the kinetic energy equation, which states KE = massv^2 (where v is the velocity of the mass). In both equations, the energy is expressed by Joules. It is almost as if the energy equivalent of a gram of matter is the kinetic energy of that mass going at c. That is, light is a mass (a photon) going at c, and all the energy it carries, it carries in the form of its kinetic energy. This, however, this is going beyond accepted theory, and a discussion of this idea is beyond what I set out to explain.
