Full Idea
Less theoretically, an ordinal is an equivalence class of well-orderings. Formally, we say a set is 'transitive' if every member of it is a subset of it, and an ordinal is a transitive set, all of whose members are transitive.
Gist of Idea
An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive
Source
Robert S. Wolf (A Tour through Mathematical Logic [2005], 2.4)
Book Reference
Wolf,Robert S.: 'A Tour Through Mathematical Logic' [Carus Maths Monographs 2005], p.77
A Reaction
He glosses 'transitive' as 'every member of a member of it is a member of it'. So it's membership all the way down. This is the von Neumann rather than the Zermelo approach (which is based on singletons).