Where does 'annulment' fit into things? Is that when it is decided to just pretend the marriage never existed in the first place.

Yes. In the Catholic Church, a "declaration of nullity" was nearly a loophole to not being able to get divorced. Basically, there were certain preconditions that were assumed to hold when getting married, and if it turns out any of those preconditions did not actually obtain, then the marriage never actually happened. For example, it is assumed that the couple wants to have children; if it turns out that one party never intended to have children, that can be grounds for a declaration of nullity.

Several legal jurisdictions have adopted this idea, but it makes little sense when one can just get divorced and there are not strict preconditions for marriage.

Wikipedia: Annulment