Answer: It's never specified
Oddly, Peter's college is never specified, nor is he ever shown going to classes. In the comics, Peter attended the fictional Empire State University, but there's no mention of that school in the film. There's a Columbia sticker (or is it a magnet?) on the side of the refrigerator in Peter and Harry's apartment, but if either of them actually attends classes there, it's never stated outright by anyone in the movie. Aunt May, while in the hospital, complains that Peter is doing too much, what with 'college and a job,' but were it not for that line, we'd have no indication that Peter goes to college at all!
Empire State University is formally identified as Peter's school in 'Spider-Man 2.'