What is the minimum number of moves, counting both sides, in which a checkmate can be made?
The precise answer to your question is four half-moves. The only checkmate possible in two moves is by Black -
1.f3 e5 2.g4 Qh4# (This mate is known as Fool's Mate). Slight variations may occur (in White's move order or in the distance Black's e-pawn or White's f-pawn is advanced):
1.g4 e5 2.f3 Qh4#
1.f3 e6 2.g4 Qh4#
1.g4 e6 2.f3 Qh4#
1.g4 e5 2.f4 Qh4#
1.f4 e5 2.g4 Qh4#
1.g4 e6 2.f4 Qh4#
1.f4 e6 2.g4 Qh4#
If White is to mate, a minimum of five half-moves (three moves) is required, for example
1.e4 f6 2.d4 g5 3.Qh5# (with similar variations possible).