One of the first true cooperative games that I played was Arkham Horror. When this game came out our gaming group just couldn't get enough of it. We played it over and over and over. Eventually we just got burned out on it. It is a game than when it isn't your turn there isn't a whole lot to do. This fact is compounded by the more players you have. If you have six people playing this game it can be a LONG time before it comes back around to your turn.
It seems that the horror genre is a pretty popular setting for this type of game as Betrayal at House on the Hill also fits the bill - sort of. This game starts out with all the players being on one side, you move around a house, gather equipment, explore, and try to survive. Once you have met a certain set of conditions the roles change - one player becomes the antoganist, and the rest of the players try to continue to beat the game. This almost doesn't count as what I would call a true cooperative game, but since you start the game as such and your roles change late I am going to let it slide.
This game started off being really fun. None of the players knew the scenarios going into it, the map is random every time you play, and when you split roles the antagonist doesn't know the other players victory conditions, and the heroes don't know the antagonists victory conditions - it makes for a real neat mechanic. The bad part about this game is that it can go quite long, has some rules issues, and you just get kind of tired of it after a while. It is worth having in your collection, but you wont be playing it every week. I just looked up this game on Board Game Geek and it looks like it is getting an updated reprint - hopefully they fix the rules issues with this game.
A good example of a game that has programed monsterser is Warhammer Quest. This is a long out of print game from Games Workshop that pits a party of adventurers against a randomly created dungeon. This game can be played with a GM overseeing the monsters, but it also plays just fine without one. The monsters will normally attack the closest player and you go from there. We were lucky enough to have most of the pieces to this game and a bit of internet sleuthing allowed us to print up the missing bits we didn't have. This game is brutally hard. You travel through a dungeon killing monsters, gaining treasure, and getting xp, and then you have to fight your way back to town to spend it. This game feels quite a bit like descent, but it is much more deadly - you just don't get to respawn back at town if you die - you are dead.
Another popular twist on the coop genre is introducing a traitor to the party. This role is usually determined in secret. Games that have this mechanic are Shadows over Camelot, Battlestar Galactica, and a few others. I haven't played any of these myself, but a few other in my gaming group have and they really enjoyed them.
All the above games are fun, but the current king of the castle amongst cooperative games is hands down Pandemic. This game is a blast. It is crazy brutal, but oh so fun. In this game you are playing a team of experts that are trying to stop a global outbreak. The board plays against you in a very pre-programmed way, but every time you play it is random. It took us about a dozen plays of this game before we beat it on the easiest setup. Since then we have beaten it a few more times, but we by no means have it on lock. You can increase the difficulty of the game by adding more outbreak cards to the deck and a new expansion makes the game even harder. It blows my mind thinking about how you would be this thing at the hardest difficulty.
If you haven't played Pandemic yet you really need to give it a try. It is easy to pick up and plays fast - which is good because you are going to lose, and you will want to play again.