What sort of platforms and genres to people primarily play? FPS and strategy (real-time or turn-based) tend to be the short-term, network-play standards. For long-term, there's MMOs, but I don't think anybody could be convinced to subscribe to one for this.

I do most of my gaming on the PC, so I'll make my list here, and leave it to someone else to offer console options.

Popular/recommended retail multiplayer games:
Team Fortress 2
Street Fighter IV
Command & Conquer (any part of the series)
Left 4 Dead (and L4D2)
Civilization IV

Good, cheap/almost-free games:
Savage 2
any Source mod (free if you already have a Source Engine game; Suicide Survival, Goldeneye Source, NEOTOKYO, Synergy)

Entirely free:
Warsow
Soldat


Post more games, or which of these you'd be up for playing, and we can get this together!