Free co-op games that aren't first person shooters and don't suck?

Alien Swarm

Barbie.

decent not free would be magicka.

League of Legends

if non-free then magicka (be warned that the game dislikes some computers at random...)

Little Fighter 2.

Realm of the mad god

Airmech?

Not Free: Dungeon Defenders! Its hell of a lot fun!

And if you like story driven coop games, I can recommend Resident Evil 5.. Singleplayer it was a little disappointing, but together (in the same room would be best) it was one of the greatest co-op-experiences I´ve ever had.

The only free one that comes to mind right now would be Alien Swarm, but I´m sure there are more.

Streets of Rage Remake V5

A fan game of high quality, easily worth playing. I think it is limited to local multiplayer though. Hope you have gamepads.

WoW Trial with friends

Spiral Knights
Micro Volts
ones already mentioned
just look on steam free to play list.

Magicka isn't bad, but is is in a genre of its own, if everyone is into it, and dicking around and having a good time, it can be great. If not it can get painful quickly.

Dungeon Defenders is pretty good if you like tower defense, but has grown expensive with the sheer quantity of DLC. It can get grindy when/if you hit endgame as well.

On the Free to Play Co-op steam lister without shooters, there is also -

War of the immortals/Rise of Immortals - ARTS/MOBA, not as good as league or DotA2.

Maple Story - MMO 2D action RPG, not... the greatest game out there, it takes a lot of time investment before it really gets fun.

Vindictus - MMO 3D action RPG, very good for its time, pretty solid if you just want to burn some time. It can also take a while before getting all that good though.

Realm of the Mad God - MMO 2D Arcade bullet hell. It isn't bad for what it is, just a question of if you're into that sort of thing. The death mechanics may be frustrating as a co-op experience, I'm not sure, haven't tried it co-op.

World of Battles - RTS with progression mechanics outside of games. Very similar to tabletop Warhammer games, cool if you want a free RTS and don't like Age of Empires Online.

Age of Empires online - MMO RTS. Feels a ton like Age of Empires 2, more so than AoE3. Genuinely pretty fun, SOME quests and whatnot are co-op capable. To do some of the stuff you do need to buy some DLC though.

Rusty Hearts - MMO Small Section Instance Action RPG. Maybe just more of a JARPGMMO? Not bad, but probably one of the worse games here.

Stronghold Kingdoms - This is a bad looking spreadsheet MMO RTS.

Fallen Earth - Haven't tried it, but it sounded and looks interesting, starts moving into the shooter genre again though.

Star Trek Online - It has some shooter sections, but is more of an MMO with some pretty interesting ship combat. Probably the game that surprised me most in 2011/2012. Very fun if you have a buddy. You don't have to like Star Trek, you just have to not hate it.

Champions Online - Pretty fun at least for a little bit. MMO. Superheros n'stuff. Not much else to say.

Spiral nights - Dungeon grinder MMO, super grindy, not very difficult, but some people seem to get addicted. I don't get it I guess.

Forsaken World - Another run of the mill MMO, couldn't stand the controls or general feel. Nothing about it really grabbed me.

Pandora Saga, and Puzzle Pirates I know nothing about.

DC Universe online (f2p mmo)

Titan quest gold (not free)

Saints Row 2 or Saints Row The Third (I think sr2 is much more better, but bad optimized. And they are not free.)

Ghost Recon Online is free. Its a 3rd person shooter, played it with a few mates and we had fun being tactical and stuff.

Team Fortress 2 Mann VS Machine.

Rayman Origins, though it's only local co-op.
Splinter Cell, because third person != first person. And it's very good co-op.
Lara Croft and the Guardian of Light for the same reason as Splinter Cell.
Trine 2.
Shank 2.
There're many more, but these are what I would recomend.

Future Cop L.A.P.D

Zombie Grinder.

most free games suck

Lara Croft and the Guardian of Light

Vindictus. I found that out of all the MMO's that's my favorite because of the actual action and not "point-and-click". I'm really waiting for Phantasy Star Online 2 to come out though because I can play on my PC and my Vita ;).

Oh and I played the Japanese Phantasy Star Online 2. Beautiful game super fun. I just can't read Japanese so I stopped after like level 12 or 13 because having to look up guides for the quests kinda sucks.

I can't think of free ones other than ones already mentioned. For non-free, Gundemonium series allows co-op. I heard DoW II and expansions let you play campaign co-op, but I never tried myself.

dota 2 XD