Network play with only 2 machines is a bit limited if you can only have 2 players.
Quake 3 Arena is great fun and will run on most reasonable PCs, and it has the advantage that you can add 'bots' to the teams. ie 6 on a side, 1 human and 5 bots, or humans versus bots, or whatever. There is also an online gaming community.
Assuming you like shoot-em-ups.