Why is the equation $E=mc^2$? The equation $E=mc^2$ never made any sense to me.
c is a constant (speed of light), therefore c squared is also a constant. We're not specifying any units so surely the equation should be reduced to this:
e = m
Why include the "c squared" part?
 A: The equation $E=m$ is only valid in one particular class of systems of units. Namely, whenever your system of units is constructed such that the speed of light $c$ is unitless and $c=1$, then $E=m$ is valid, because, in those systems, mass and energy are expressed in the same units.
On the other hand, the equation $E=mc^2$ is valid for all systems of units, because it accounts for the case where $c$ has units and also accounts for systems of units where the numerical value of $c$ is not equal to $1$.
For example, if you were using SI units, where energy is specified in joules (meaning kg m$^2$/s$^2$) and where mass is specified in kg, then $E=m$ wouldn't make any sense at all - the units don't match up! It would be like saying "$3$ seconds is the same as $5$ liters": there are some contexts in which that would make sense, but only after you had established some kind of direct conversion between seconds and liters. (Setting $c=1$ establishes just such a conversion, which is why $E=m$ works and why mass and energy are specified in the same units for any system where $c=1$.) In contrast, $E=mc^2$ works perfectly, because a mass (kg) multiplied by a squared speed (m$^2$/s$^2$) gives you precisely the right units for energy, and so the two sides can be sensibly equated. 
$E=mc^2$ is the more general expression, and $E=m$ is a special case for particular systems of units. 
That said, $E=mc^2$ (where $m$ is the rest mass of the object*) only applies to objects that aren't moving. The most general expression relating the mass and energy of a point particle moving with momentum $p$ is:
$$E=\sqrt{p^2c^2+m^2c^4}$$

*Historically, there is a concept called relativistic mass which is defined differently, but this is not often taught in the modern era. Nowadays, we usually think of mass in relativity as synonymous with rest mass, but we still use the term "rest mass" to avoid any ambiguity.
A: Einstein's original formulation was Energy (in ergs) = mass (in grams) x c (speed of light) squared in centimetres per second. Ergs have fallen out of favour, so they probably use different units nowadays. This site keeps correcting my English spelling, not withstanding the fact that the true authentic English is from England, just as real champagne is from France.
