set theory - Axiom of Choice and Order Types

Order equivalence is an equivalence relations on ordered sets, not on sets. It is just the isomorphism relation on ordered structures. An ordered structure is a set, together with an order.

The Axiom of Choice says that every set has a well-order. Since the order-types of well-orders are well-ordered (given any two, one of them is uniquely isomorphic to a unique initial segment of the other), it follows under AC that for every set, we can associate to it the smallest order-type of a well-order on that set. This is called the cardinality of the set.

There is another more general concept of cardinality, which does not rely on AC or on orderings at all, and it is just the equinumerosity class of the set.