White starts and the first player to promote a pawn, capture all his opponent's pawns, or run his opponent out of legal moves, wins the game.
With one pawn each it's trivial, with two pawns each it's easy, and with three pawns each, the bright child will soon realise that a reflecting strategy wins for Black: 1.a4 is met by 1...c5! or 1.c3 by 1...a6! (but 1.a4 a5? 2.c3 c6 3.b4! wins for White!) Once you get to four or more pawns, however, the correct strategy becomes very complex, and playing with all eight is almost as difficult as chess itself.
The diagram position (above, from Ilyin-Zhenevsky-Alekhine, 1920) shows that the game has real practical applications too.
It's Black's move, and he can only save the game by running White out of Q-side pawn moves. If he runs out of moves himself, then his king must move, losing the h-pawn.
There is only one move that leads to a draw. 1...d5 loses to 2.exd5 cxd5 3.a3; 1... c5 loses to 2.b4 c4 3.a4 and 1...b5 loses to 2.b4 c5 3.e5! Even 1...b6 loses (work it out for yourself).
Alekhine played 1...a5!, and the game was drawn after 2.c4 b5 3.cxb5 cxb5. Now, after 4.b3 b4 or 4.a3 a4, White's king must retreat, and Black's can finally step back safely. Alekhine, of course, had it all worked out even before going into the king and pawn endgame.