The National Science Foundation sponsors a number of summer research internships for undergraduate mathematics majors.
These programs typically last 8 to 10 weeks, and during that time students are exposed to a new research topic and given a problem of their own to work on. Wellesley students have been regular participators in REUs, and their experiences have been overwhelming positive.
If you are interested in participating in an REU for a given summer, you should speak to a faculty advisor as soon as possible in the preceding school year. Applications are often due in the winter or spring, though one should usually start thinking about her application well in advance of that time.
Available REU programs typically vary from year to year, though the AMS provides a list of REU programs (beware: this list isn't always up to date!). If you are preparing an REU application, you would be well-served to read the following guide to competitive REU applications composed by the Grand Valley State University REU program.