You can join other players to create a crew, multiple crews join up to create a faction.
Faction can be part of the 3 Orgs or 2 Splinter Orgs:

Zion (E Pluribus Neo) - Machine (Cypherites) - Merovingian

Only the main 'Orgs' (Orginization) have controllers you can do missions for, they will build your reputation and decrease it for the other 2 main Orgs.  You can do missions for the enemy but you will lose rep with your org and wont be able to do the weekly critical missions (story related) or enter the white hallways to go to that Orgs special area...

...I hope this helped as I gave myself a headache trying to make it understandable ^_^