Question

whats another equivalence class over the integers besides mod n

Answer 1

... its like i can almost make sense of the question, but it still eludes me :)

Answer 2

i tend to get lost in the terminology

Answer 3

there are a lot of equivalence relations, but "mod n" is one of the few interesting ones that has amazing applications.

The creation of the Integers is an equivalence relation itself, acting on the cartesian product of the natural numbers with itself. You would say something like:
\[(a,b),(c,d)\in \mathbb N \times\mathbb N \]\[(a,b)\sim(c,d) \iff a+d = b+c\]

Then there are equivalence relations of sets of objects that arent numbers. Like matrices for example.