Summer (Research) Opportunities/Internships
If you are thinking about a career in mathematics (where a "career" is not strictly-speaking "academia"), you should probably consider a summer program. Here are some options:
ACADEMIA/RESEARCH RELATED
Research Experiences for Undergraduates (REUs). Colloquially, "REU" implies NSF funding. This means you need to be an American citizen or permanent resident in order to apply (of course, you should still email any programs that interest you and ask about related opportunities). This also means there's a fair amount of competition. Typically 6-12 students annually are accepted from a national applicant pool. The areas of research vary from year to year, and from institution to institution. Applications are usually due between February 1 and February 21 of a given year but that too will vary with the program.
Personal plug: Along with Jeremy Rouse at Wake Forest, I am running a number theory research experience for undergraduates (REU).
The NSF's list of funded REUs can be found here.
The AMS has a separate list (there's a lot of overlap, though).
Another research opportunity is the Park City Math Institute (associated with the Institute for Advanced Study).
Carleton College has a summer program specifically for female mathematics students in their first or second year of undergrad. This is not an REU, yet is still NSF-funded. Therefore, you need to be an American citizen. [No funding this year.]
For women and other minorities who intend to proceed to graduate school and/or have just completed their first year of graduate school: consider the Enhancing Diversity in Graduate Education program (EDGE). You do have to be a US citizen.
"MATH CAMP" COUNSELOR
Summer Mathematics Programs for high school students. If you like the idea of inspiring the next generation of mathematicians, and/or the general idea of an (intense) math summer camp, you should apply to be a counselor! Here are some notable summer mathematics programs for high school students which prefer undergraduate counselors:
The Program in Mathematics for Young Scientists (PROMYS) at Boston University
NON-ACADEMIA RELATED
Internships at the NSA. The National Security Agency loves mathematicians! If you're thinking about a job outside academia and/or in government, you should really consider applying. These applications will be due VERY EARLY (quite frequently, you want your application submitted in the fall semester before the summer in question--security clearances take time!). Again, you typically have to be an American citizen or US resident in order to apply.
The Institute for Pure and Applied Mathematics (IPAM). This group sponsors various industry-based summer programs for undergraduates. Also worthy of note: historically they are welcome to both US and foreign applicants.