Research Experience for Undergraduates (REUs) are summer programs sponsored by the National Science Foundation (NSF). REUs usually consist of two parts: intensive study of topics through lecture and interaction, and student research on a question/questions. Travel costs are paid for as well as room and board. A stipend is given to participants. These are all available on a competitive basis. Students that participate in REUs often present their research at national meetings.
Both AMS and the NSF keep current lists of math REUs. Most deadlines for the summer programs are in February and March.
Not sure what it would be like to participate in an REU? Take a look at this article from our archives: Is an REU for you?