There are a lot of different kinds of mathematical experiences students can have over the summer. Many students engage in research either on- or off-campus, others attend programs where they essentially take classes, and a few find internships that have a mathematical sciences flavor.
Research Experience for Undergraduates Many colleges and universities have REU programs sponsored by the National Science Foundation. Students spend 6-10 weeks working on a research project given to them by the faculty participating in the REU. Usually students spend a couple of weeks at the beginning of the program learning their topic area, then work in groups on the research. Some REU programs are designed for rising seniors who are very serious mathematics students, while others are designed for rising juniors and even younger students who do not yet have a lot of mathematics courses under their belts. Unfortunately, you'll need to wade through a lot of links to find the right program for you.
- NSF Sponsored REUs: This is a list of schools currently receiving funding for REU programs from the NSF.