It's been jokingly called the "Not Important Tournament" or the "Not Invited Tournament", even the "Nobody's Interested Tournament", but the National Invitational Tournament or NIT is the oldest continuing NCAA basketball tournament. According to the wikipedia, the first NIT champion was the Temple Owls in 1937.
The NIT chooses the best teams that have not qualified for the NCAA tournament. There is a field of 32. The games are televised on ESPN.
The tournament was started by the Metropolitan Basketball Writer's Association of New York City. Originally, the entire tournament was played in Madison Square Garden, but since 1977, the preliminary rounds have been played on the campuses of the top seeded teams. The Semi-Finals and Finals are still played at the Garden.