C3 was the short name of the Chrysler Comprehensive Compensation project, a payroll project at Chrysler which has since become famous as the 'birth project' of Extreme Programming.
The project was an attempt to replace the many legacy COBOL payroll systems that existed. I was involved as a consultant in a number of ways starting in 1993. The project began some serious development work, in Smalltalk, in 1995, but was not able to reach a stable state and was restarted under Kent Beck's lead in 1996. It was this rebooted project that first pulled together all the practices that became known as Extreme Programming (although Kent had used similar approaches on previous projects.)
The software went live in 1997 paying around ten thousand people. The project continued, intending to take on a larger proportion of the payroll but new development was stopped in 1999. Later on the existing live system was reverted back to the COBOL (although last I heard it still paid a few people who could not be handled with the existing software). I also heard that the intention is to use an ERP system to handle payroll in the future.
C3's early success was a great inspiration for XP, and C3 had an important role in blazing the trail for XP. By the time the project was canceled, enough other projects had replicated C3's early success to continue XP's development. C3's cancellation, however, also proves that XP is no guarantee of success.
Many people have tried to analyze C3's success and cancellation. I haven't seen any analysis based on much knowledge of the facts. I've always felt that a full description really should come from people on the team full time, and so haven't written anything myself. I was only involved part-time and only until late in 1996. In particular the entry in Wikipedia is misleading and incomplete, much of its comments seem to be based on a paper from a determined XP critic whose sources are unclear. Certainly its comments on performance are a misleading interpretation of material in my Refactoring book.
Many of the C3 team have gone on to further XP work both in and beyond Chrysler. One group continues to use XP extensively on a project notable for being a VeryLowDefectProject.