Nick has it mostly right ... at this point, very few carriers are developing or maintaining their own proprietary systems. Most of the systems originated internally but have since been sold off (like American did with Sabre). Today, many carriers use components from several vendors to fulfill different parts of their operations (dispatch, checkin, maintenance, weight/balance, reaccommodation, inventory, revenue management, etc.).
Sabre, Amadeus, and Worldspan are all in this business as well as their traditional travel distribution business (via agents) and have developed easy (or at least easier) to use graphical front-ends, but there are a bunch of smaller providers that sell a suite of services as well as graphical front-ends to the legacy systems.
As DeltaGuy mentions, Delta still has a lot of internal development on Deltamatic (their platform) which shares infrastructure with Worldspan, IIRC. Northwest is fully hosted in Worldspan as well. American and US Airways are in Sabre, while Continental and America West are in Shares, and United is in Apollo.