Most people are very familiar with games that promote competition, the majority of games do. Whether they are sporting events, tabletop games or card games: cooperative games are more elusive.
While some games have a cooperative element mixed in (games with teams, for example), they are often still competitive. In team games, the team members themselves obviously are cooperating, but only in the greater context of competing against other teams. More than few team building activities work this way.
There are some games that are purely cooperative, where all players are working together for a common goal. (yes, there are some ‘gamers’ out there who love competition and would not consider cooperative games to be games)
A great list of games considered Cooperative can be found on http://www.boardgamegeek.com(search using advanced options)
I love to design games that have the illusion of being competitive, or at least start that way, but cleverly become cooperative as they proceed. As a game designer, I consider the balance between cooperation and competition to be very interesting. There are countless ways to do this, but it can be a real challenge!