The main MP elements you can access are Crucible (PvP, usually in teams), Strikes (3-player story missions that take 15-20 minutes), and Gambit (PvE with a little PvP mixed in). Nightfall is a weekly strike with better rewards, but usually tougher. The game wants you to do that with players you know, although you can be paired with randoms if you're willing to wait. I have no idea why they do that - the missions are just hard strikes. You can also do Raids, which are large story missions for big groups. Those are massive and surprisingly unlike the entire rest of the game. They're VERY heavy on coordination and communication. There are also public activities scattered throughout the maps, too. Those involve a stated objective and if certain conditions are met, it'll turn "Heroic" which nets better rewards.