In set theory, the ordinal numbers are objects that represent the order types of well-ordered sets. Is there a similar class of objects that represent the order types of well-founded sets? If so, are there any good resources on them?
The ordinals are extremely nice because they have a Cantor-Bernstein like theorem. If $\alpha\leq\beta$ and $\beta\leq\alpha$ then $\alpha=\beta$.
This is not true for general well-founded orders, as Brian's answer here shows. So for general well-founded orders embeddings do not make a partial order on isomorphism classes.
Generally speaking, though, if you just want to have representatives you can always just choose a representative from every isomorphism class, you can even make sure that these are sets and their relation is $\in$ relation.
The Mostowski collapse lemma says that if $(A,R)$ is well-founded and extensional then there exists a unique set $M$ such that $(A,R)\cong(M,\in)$, which begins to sound a bit more like what you are looking for. I suggest that you try and play with things from there.