Participating in a Research Experience for Undergraduates (REU) summer program is a great way to get involved in mathematical research. The NSF and AMS maintain lists of math REUs around the country; Yale also runs SUMRY, an internal REU, every summer. REU applications are typically due in early February. One thing to keep in mind is that external REUs are often extremely competitive and even harder to get into than graduate schools.
Outside of REUs, students sometimes work on research problems directly with a professor. Yale has a number of fellowships and grants which may be helpful in funding individual summer research.