The shortest chess game between two people is often referred to as the "Miniature Game" and was played between Byrne and Fisher in 1956. The game lasted only 21 moves and ended with Byrne resigning.
Here's the game:
1. Nf3 Nf6
2. g3 g6
3. Bg2 Bg7
4. O-O O-O
5. d3 d6
6. Nbd2 Nbd7
7. e4 e5
8. exd5 Nxd5
9. Nc4 Nb6
10. Ne3 Qe7
11. Qe2 a5
12. a4 Nc4
13. Nxc4 Qxe4+
14. Qxe4 Nxe4
15. Bxe4 Bxe4
16. Nd2 Bxg2
17. Kxg2 f5
18. f3 e4
19. fxe4 fxe4
20. Nf3 e3
21. 1-0 (Byrne resigns)
This game is often cited as one of the shortest games between two grandmasters.