The wife and I play lots of different co-ops together. Mostly those with a campaign system, but we've played plenty of others. Here's some quick thoughts on those I have played. Let me know if you want more details on any of them.
Pandemic - Save the world from nasty diseases. One of the first modern co-op games. At this point I think of it as being an average game at best. Gets pretty repetitive and can suffer from one Alpha player running everything. Best with 4.
Space Alert - Crew a spaceship on missions scouting dangerous sectors of space in the SS Sitting Duck. I love this game, but it is damn stressful. Introduces a time mechanic that forces fast decision making. Really fun with the right group, but not for everyone. Still fun even when you are losing badly. Best with 4, but 5 players works too. The campaign system is a fun add. Designed by Vlaada Chvatil who is easily my favorite game designer.
Forbidden Island - Explore and island and grab treasure before it falls into the sea. A good co-op for non-gamers as it is really light. Don't own this one, but have played a few games.
Lord of the Rings LCG - Build decks and lead heroes through quests in Middle Earth. Mechanically this one is great and there are tons of cards available at this point. They've released campaign expansions that follow the books, but most of the game explores other quests and parts of Middle Earth. My wife likes to play, but has no interest in the deck building so I handle that. Best with 2, but works fine with up to 4.
Kingdom Death: Monster - Can you build a settlement that can survive a nightmare world by hunting monsters? Super expensive and really dark, but the game is mechanically a lot of fun and easy to get invested in as a player. Best with 3-4 players. Somewhere between a miniature game and board game. The game we are currently playing the most.
Red November - Save a bunch of drunken gnomes on a badly damaged submarine. Fun for the theme, mechanically its all about pressing your luck. We have a really low win rate. Supports up to 8 players and I wouldn't play with fewer than 5.
Heroes of the Aturi Cluster - This is a mod for the X-Wing Miniatures Game that turns it into a mission based co-op campaign. You need a ton of ships to make it work, but the payoff is absolutely fantastic. Best with 4, but would work with 2-6.
Mice and Mystics - Turn into a mouse, grab some friends, and save the kingdom from an evil sorceress. Mechanically fun, but simple enough for kids to enjoy it too. Good storytelling throughout the game. 3-4 players.
Legends of the Alliance (Imperial Assault) - The recently released app for Imperial Assault allows a full co=op mode. We love it. Best with 3-4 players.
I'm sure I've played others, but that's what I can think of right now. As I said, let me know if you want more details on any of them.