Probably my favourite definition in
set theory, if not the whole of mathematics, and the only
recursive definition I know of. A set A is said to be hereditarily finite if and only if A is finite, and all the members of A are hereditarily finite.
What this means in effect is that A is a finite set, the members of A are finite sets, and their members are finite sets, and so on.
There is actually a set containing the hereditarily finite sets and nothing else, which is nice because there's no set containing all finite sets (since if there were such a set K, then it would have to contain {K}, and that's trouble). Start with the empty set - call this V0. Then take the power set of this (ie. the set {{}} of all subsets of the empty set) and call it V1. Take the power set of that, call it V2. Keep going like this, and you get a sequence: V0, V1, V2, V3, ... Now take the union of all these sets, and you get a set called Vomega, and it turns out that this is the set you want.