History of Mathematics: Winning strategies

This column's topic will be a theorem in the mathematical theory of games. It concerns what could be thought of the simplest possible type of game we can imagine.
 
It is played by 2 players who move alternately.
 
It is a game of perfect information; there are no hidden data such as would occur with a card or dice game; both players are fully aware at all times of the state of the game.
 
The game ends within a finite number of moves.
 
It is impossible to have a drawn game; one or the other of the players must win.