The following might be more appropriate as a comment to sleepless in beantown's answer instead of an answer, but for some reason I am not able to comment.

The following website of the AMS contains numerous links to Research Experience for Undergraduates (REU) programs

http://www.ams.org/programs/students/undergrad/emp-reu

If I understand correctly, these REU programs are funded by the NSF to offer undergraduate students (not necessarily from the institution that is hosting the program) an opportunity to do research, under supervison, over the summer. (Since there was some debate what "research" should mean, I add that here "research" means, or at least can mean [and not only rarely], something that in the end is published in well-established mathematical research journals.)

Also, there is a somewhat recently founded journal Involve specifically dedicated to "showcasing and encouraging high quality mathematical research involving students (at all levels)"

http://pjm.math.berkeley.edu/involve/about/journal/about.html