As some people remarked, you don't need to know about ordinals at all in order to understand a proof of Zorn's lemma. However, I think that ordinals make an incredibly lovely concept, and show a particularly great generalization of something we all know very well (the natural numbers and induction), so it is worth learning a little about them.
What do you need for understanding the classical proof of Zorn's lemma?
- What is a well-ordering, what are successor and limit points of a well-order.
- Transfinite induction/recursion on general well-orderings.
- Every two well-orders are comparable in the relation "isomorphic to an initial segment".
- If there is an isomorphism between two well-ordered sets, then there is exactly one.
- What is the von Neumann ordinal assignment, and why it's a good thing (in particular how $<$ is really $\in$, and $\leq$ is really $\subseteq$).
- Hartogs and Lindenbaum theorems.
Now it should be easy to understand the majority of the proofs of Zorn's lemma from the axiom of choice.
It should suffice for understanding the usual proof of the well-ordering theorem.
Of course, that you can decide that you want to do things a bit differently, and use some other principle and not the axiom of choice directly. For example "Every partial order has a maximal chain", then the proof of Zorn's lemma is trivial, since an upper bound of a maximal chain is a maximal element!