I understand that an equivalence relation is a set that is reflexsive, symmetric, and transitive. I don't quite understand equivalence classes though.
For example:
What would the equivalence class be of the equivalence relation {(0.0),(1,1),(2,2),(3,3)} if the (original?) set is {0,1,2,3}?
On
The equivalence relation is a set of pairs of equivalent things. In your example each element is equivalent to only itself so the equivalence classes are $\{0\},\{1\},\{2\},\{3\}$. If you add $\{(1,0),(0,1)\}$ (you need both pairs to have symmetry) to your equivalence relation then you'll have classes $\{0,1\},\{2\},\{3\}$. Generally your initial set will break up into a disjoint union of equivalence classes whenever you have an equivalence relation.
Consider an equivalence relation on the set $S=\{1,2,3,4,5,6\}$ which is given by $$R=\{(1,1),(2,2),(3,3),(4,4),(5,5),(6,6),(1,3),(1,6),(6,1),(6,3),(3,1),(3,6),(2,4),(4,2)\}.$$ You can verify that $R$ is an equivalence relation on $S$. Now, we define an equivalence class of $s\in S$ as $$[s]=\{x\in S : x\ R\ s\}$$ So, what elements in $S$ are related to $6?$ They are $\{1,3,6\}.$ Using above notation we have $$[6]=\{1,3,6\}$$ The set $\{1,3,6\}$ is called an equivalence class of $6$. I'll leave the rest to you to see if you can find all the other equivalence classes.