OpenFeint started as a side project of Jason Citron and Danielle Casley, the 2-man development team behind the Aurora Feint series of games that originated on the iPhone. It was started when there was no unifying social platform on the iPhone, along with several competing services, such as ngmoco's Plus network, and Chillingo's Crystal.
Today, OpenFeint has almost 20 million users, which is almost 25% of the current market share of iPhone and iPod Touch owners. The service is completely free to users, and only costs money to developers who charge for their games.
Post Apple's Game Center
Apple's Game Center released on September 8th, 2010, and while many developers have been updating their apps for Game Center support, there are still many games that have not been updated. This is most likely a combination of lack of developer support, as well as conscious developer decisions to stick with Game Center.