I'm just not really sure what well-definition implies. I think I get it, but I could use a little feedback.
Yes, that is correct. Well-defined means if you pick two things from the same class and multiply (or do any other operation) you will always get the same answer - it doesn't matter which element from the class you pick. So, whenever you see something whose elements are of the form $\displaystyle [a]$ or $\displaystyle a+I$ or $\displaystyle aN$ where $\displaystyle I$ or $\displaystyle N$ are some sort of algebraic structure (for instance, $\displaystyle I$ an ideal or a ring, $\displaystyle N$ a normal subgroup of a group) you would need to take two elements and see if they are always mapped to the same thing.
For instance, let $\displaystyle S=\{0, 1, 2, 3, \ldots, n\}$ and let $\displaystyle T = \{[x]:[a] = [b] \Leftrightarrow n|(a-b)\}$ be the set of equivalence classes of the integers modulo n, and again let the operation be addition.
Can you show that $\displaystyle \varphi: [x] \mapsto x \text{ mod } n$ is well-defined?