This National Science Foundation program is designed to give undergraduates, especially those from smaller schools, a chance to gain real research experience for a summer. Personally I participated in one official REU and one program modeling on REUs. I learned a lot (and they were tons of fun!). The best part is not the specific topic you research, but the opportunity to learn how to be a researcher.
Most of the applications are due in February. Check out the the official NSF REU website for the latest details.
When you are ready to apply, go here to search for programs of REUs in various subjects. Also, search the internet for other research opportunities; Harvard has a nice list of research programs for undergrads. For more detailed tips on applications, I recommend this site.
If you want to get an idea of what an REU is like, here are some interviews of past Math REU participants. And also keep in mind these research tips for undergrads if you do get an REU.
N 2 Infinity and Beyond 