Two years ago I worked on the coop part of Tintin the video game, and i made a Pecha Kucha to sum up all the different kind of coop mode that exist in games. Let's convert it to an article!
A little bit of history for the first two slides: Double Dragon (1987) & Contra (1987) wasn't maybe the first coop games ever, but they are the first one which have had a great success.
There would never have been LAN parties without the Doom multiplayer LAN mode (1993).
Now let's start talking about the big categories of coop games: the first one is the solo game played in coop (same levels, same 3C, ...) like in New Mario Bros Wii (2009).
The second one is the solo that can be played in coop but with a different tuning (more enemies, more life, ...) like in Metal Slug (1996).
In Little Big Planet (2008), there are special zones that can be played or reached with the accurate number of players. The big issue with this kind of design is that a solo player can't complete the game at 100%.
In Gears of War (2006) there are several coop zones in the walkthrough where the players must take different paths. In solo, the other players are replaced by AI.
In Lego Batman (2008), the game is designed mainly for the coop with each character having its own special ability needed to go trough the levels. In solo the player can swap between the two characters. The only design limitation: you can't design puzzles that ask the use of the two characters' special abilities at the same time.
Lara Croft and the Guardian of Light (2010) is almost designed in the same way, except that the levels are a little bit different in solo and coop (for instance, a crate replace the second player in a two ground switches puzzle). The only flaw, the level must be reloaded when a second player opt in the game.
Another way to add coop gameplay is to add special coop levels in the walkthrough only playable by two players like in World Of Illusion (1992). In solo you don't play these levels.
You can also design a coop game and add bots when you play in solo (Left for Dead 2008).
Or adding a specific coop campaign like in Splinter Cell: Chaos Theory (2005).
Now let's talk about the camera because designing a coop game implies a particular camera behavior. We will talk about the way you can respawn in the game too, and death conditions.
In Final Fight (1989) the players can't go out of screen. The camera stops at particular position and is unlocked when the players have defeat all the ennemis.
In New Mario Bros Wii (2009) the camera keep the screen edges collisions but zoom out when the players are too far from each others. Then the camera zoom in with a delay to avoid the "vomiting effect". The first player drag the camera with him. If you die you respawn in a bubble and need help from the other players to go back in game.
In Lara Croft (2010) the screen edges collide too and you press start to respawn near the other player.
In Donkey Kong Returns (2010) the screen edges doesn't collide. When a player is out of screen he is automatically respawned near the other player after a timer displayed on screen. When you die you respawn in a floating barrel.
In Little Big Planet (2008) there's also no screen edges collisions, but when you stay too long out of screen you die. Then you respawn at checkpoints (so the respawn positions are controled by the designers).
In Army of Two (2008), because it's mainly a coop game, the death of only one player leads to a game over.
In Kirby's Epic Yarn (2010) you have infinite lives (no lives in fact). The camera & respawn system is similar to New Mario Bros Wii. There's a game over only when the two players dies at the same time.
Up to now I talk about cooperative games & gameplay. But you can also add some competition in cooperative games like in Little Big Planet with the final level score. It's similar in The Legend of Zelda: Four Swords Adventures (2004). But be carefull with the balance between cooperation and competition. If you add too much competition in your coop game, you will ruin the experience.