What is the least amount of moves it takes to checkmate in chess?

The shortest mate is called "Fool's mate" and requires the cooperation (or lack of skill) by the opponent. It requires 5 moves (3+2) for White to mate Black, but only 4 moves (2+2) for Black to mate White.

Fool's mate by White

1. E3 f6

2. * g5

3. Qh5 mate

(*any move except Q-h5 or one that blocks the diagonal for the queen)

Fool's mate by Black

1. f3 e6

2. g4 Qh4 mate