Each year a number of universities across the country offer summer research opportunities for undergraduates.  These programs, called Research Experience for Undergraduates (REU), normally include financial support.  Typically, students apply for these programs in the spring of their junior year and participate in the summer between their junior and senior years.  Interested students should consult with a faculty advisor. 

The American Mathematical Society has a list of REU programs, and the Mathematical Association of America has a list of some programs as well as meetings geared toward undergraduates