http://www.rockstargames.com/reddeadredemption/features/multiplayerscrolling to the bottom of that page gives you a description of the games. The competitve modes are player vs player up to 8 v 8. The co-op are exactly that: everyone against the computer to complete the objective
Here are links to the specific DLC pages with more info. I still haven't tried everything yet. They come in a pack on the PS store, all 3 for $20 or $10 each.
http://www.rockstargames.com/reddeadredemption/dlc/undeadThis is a blast, my son likes it better than COD zombies.
http://www.rockstargames.com/reddeadredemption/dlc/liarshttp://www.rockstargames.com/reddeadredemption/dlc/legendsThis one is free
http://www.rockstargames.com/reddeadredemption/dlc/outlaws