Why is $x^0 = 1$ except when $x = 0$? Why is any number (other than zero) to the power of zero equal to one? Please include in your answer an explanation of why $0^0$ should be undefined.
For non-zero bases and exponents, the relation $ x^a x^b = x^{a+b} $ holds. For this to make sense with an exponent of $ 0 $, $ x^0 $ needs to equal one. This gives you: 

$\displaystyle x^a \cdot 1 = x^a\cdot x^0 = x^{a+0} = x^a $

When the base is also zero, it's not possible to define a value for $0^0$ because there is no value that is consistent with all the necessary constraints. For example, $0^x = 0$ and $x^0 = 1$ for all positive $x$, and $0^0$ can't be consistent with both of these.

Another way to see that $0^0$ can't have a reasonable definition is to look at the graph of $f(x,y) = x^y$ which is discontinuous around $(0,0)$. No chosen value for $0^0$ will avoid this discontinuity.
Because when you say 'to the power of zero' it just means that you are dividing that number by itself. Therefore zero to the power of zero is zero divided by zero which is undefined.