Basic question: what does the notation $[A,B]$ mean? If $A$ and $B$ are both matrices, what is $[A,B]$? I understand that it is a commutator and that $[A,B]=AB-BA$, but since I don't know what a commutator is, none of this information is telling me much.
 A: As you note: $[A, B] = AB - BA$. Rather than spell out the way to compute it every time we refer to it, we simply call $[A, B]$ the commutator.
As you see if you explore the link above, is that commutators exist in many contexts, not just when discussing matrices. 

"In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory." 

I'd suggest you read the Wikipedia entry and particularly, note that the commutator of two elements $a$ and $b$ of a ring or an associative algebra is defined by $$ [a, b] = ab − ba. $$ 
A: It means exactly what you say. $[A,B]$ is precisely $AB - BA$, and people happen to call it the commutator.
A: Math objects which obey the commutative property of multiplication satisfy $ab=ba$, and therefore $ab-ba=0$. 
Matrix multiplication does not satisfy the commutative property so in general $AB\neq BA$, therefore $AB-BA \neq 0$. Because this form of expression comes up a lot, mathematicians invented a compact notation for it: $[A,B] \equiv AB-BA$, which is called "the commutator of $A$ and $B$."
Like Robert and mixedmath said already, there is nothing to understand here --- it is just a definition.
