Full Idea
Zermelo's Axiom of Choice asserts that for any set of non-empty sets that (pairwise) have no elements in common, then there is a set that 'simultaneously chooses' exactly one element from each set. Note that this is an existential claim.
Gist of Idea
Axiom of Choice: a set exists which chooses just one element each of any set of sets
Source
Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I)
Book Reference
Feferman,S/Feferman,A.B.: 'Alfred Tarski: life and logic' [CUP 2008], p.46
A Reaction
The Axiom is now widely accepted, after much debate in the early years. Even critics of the Axiom turn out to be relying on it.