Most video games these days would essentially have factions you would work for in the single-player campaign and as for multiplayer, you would usually be in some sort of team to fight the other team.
Title says it all. I'm just going to ask you guys what factions and teams do you support or favour in gaming? You can add in Organisations too.
Here is a list of a few of my favourites:
The United Nations Space Command - Halo Series:
The Dark Brotherhood - Most Elder Scrolls Games:
Reliable Excavation Demolition - Team Fortress 2: