Many Mathematics Departments and Universities have summer programs called "Research Experiences for Undergraduates," or REUs for short. They tend to be 6-8 week summer programs aimed at helping undergraduates become involved in mathematical research. If you are considering going to graduate school, these research experience programs are an excellent way to "test-drive" graduate school. Many such programs are funded by the National Science Foundation and provide travel expenses and stipends, at least to US citizens and permanent residents. The NSF maintains a directory of currently operating REUS:
Although details vary by individual site, applications tend to be due in February each year.
Another good resource for searching for mathematically oriented research experiences and summer programs is: