Haven't dove into the campaign yet, figured I'd give some impressions of the 4-5 hours I've spent on the multiplayer. Obviously if you don't want the modes spoiled for whatever reason, don't read this post.
All multiplayer modes are 4 or less players. Sound weird, yeah, but they're designed around that and it works.
Contractions:
As people have said, this mode is like a horde or zombies mode. You hold up in a base and try and survive waves of enemies. They get harder and harder and by the 10th wave it's damn difficult. After each wave you can venture out into the map to find cargo boxes to bring back to your base, which adds certain weapons and ammo. There's usually 2 or 3 yellow boxes which add normal weapons and ammo, and 1 red box which adds a special weapon (sniper, missile launcher, railgun, etc).
The biggest map has a mech in it, so you or a friend can grab the mech and fight against other enemies and other mechs, which are exclusive to that level. This is probably the funnest Contractions map, in my opinion, both for the mech player and the regular players.
Fucking Run:
Pretty straight forward, you run from a giant wall of death while gunning down enemies. Rounds last 1 or 2 minutes on average, you're lucky to get past 2 minutes. When the wall gets close, the screen and volume pounds and shakes and what not. It's pretty intense.
Soul King:
Soul King is pretty much deathmatch with a twist. You're all ghouls and the goal is to possess soldier bots that are fighting around the map and take out other bots/players to collect souls. When you kill a bot, they give a small amount of soul points. When you kill a player, they give a certain percentage of what points they have. The person with the most soul points is the "Soul King". The only difference between the Soul King and normal players is that normal players can see where each other are. Soul Kings are blind to where others players are, which makes it impossible to distinguish between the bots and the players.
You can only do damage while you're possessing a soldier. As a ghoul, you can only blur people's vision.
Soul Survivor:
This is probably my favorite mode. One person at the start of the round is make a ghoul, and the goal of the ghoul is to convert the other 3 players into ghouls by possessing bot soldiers around you, gunning down the players, and standing over them and holding E. The goal of the other 3 players is to survive the constant waves of bot soldiers and, of course, the ghoul.
Each game mode has 3 maps, for a total of 12 maps. Me and my friend have had a blast so far, and are looking forward to trying out the campaign tonight.