The big list:

A compilation of opportunities in mathematics is posted at the University of Nebraska at Lincoln. This is a great list which includes some of what I have below and much more.  If a link is out of date, often a Google search will take you to the right page.

Things missing from the big list:

The Mathematical Sciences UP Program  From program announcement:  "MSRI-UP is a comprehensive program for undergraduates whose primary aim is to increase the number of students from underrepresented groups in mathematics graduate programs. MSRI-UP includes summer research opportunities, mentoring, workshops on the graduate school application process, and follow-up support."

This Society for Industrial and Applied Mathematics webpage lists a whole bunch of opportunities in applied math.  Scroll down the page to find the list for undergraduates.

Programs with which I am familiar:

The Johns Hopkins University Centers for Talented Youth are a great place to gain teaching experience during the summer.

The Budapest Semesters in Mathematics program is a wonderful way to study abroad and pursue mathematics. If you'd like to learn more about the program ask me about my Fall 1994 semester in Hungary!

A great way for undergraduates to get some experience in math or science research is to do an REU. (Research Experience for Undergraduates.) Click here for a list of REU's.

The Carleton/St. Olaf College Summer Mathematics Program for Women Undergraduates is a program for first and second year women who are interested in mathematics.

The Enhancing Diversity in Graduate Education (EDGE) program is a summer program aimed at strengthening the ability of women and minority students to successfully complete graduate programs in math. Students about to enter a grad program, or those in their first year, are eligible to apply.