You must finish a session of each of the 11 game types.
I found this achievement to be mildly frustrating due to the lack of people online, although it's still very much achievable.
You don't have to score any points, or even play an entire match (I.e you can join late), but I believe you need to stay until the end once you've joined.
To check which modes you have already done; go to multiplayer, then hit RB to bring up the Friend Hub, from here select the Friends Ladder and move to Game Modes. If you've scored points in a game mode, you've done a match.
You'll want to start by finishing both tutorial sessions:
- Introductory Session 1
- Introductory Session 2
Usually there seems to be more people in the TEAM OBJECTIVES games, so your best bet is to start with those as you can gather recent players who you will invite to a private match for the more illusive FREE FOR ALL modes.
The TEAM OBJECTIVES modes are as follows:
- Artifact Assault
- Chest Capture
Once you've finished the above modes, I'd search for games in the FREE FOR ALL modes, although they are less popular.
The FREE FOR ALL modes are:
- Steal The Artifact
If you are having no luck, invite some recent players to a private match of any of the modes you've missed. Hopefully a few of them will join and you can get them out of the way.
The friend feed (which you used to identify which modes you've done) doesn't show any data from private matches, so take note of these separately.
If you do all 11 of the modes and don't get the achievement, try the SIMPLE DEATHMATCH (people have reported this doesn't count towards it, but it's not been confirmed)