In Summer 1994, I participated in the National Security Agency's Director's Summer Program (the DSP), a 12-week undergraduate math research program at the NSA headquarters at Fort Meade, MD.
The first two weeks we learned cryptographic math and basic security procedures. At the end of this orientation, many potential problems were listed. (The problems all have practical (cryptologic) origins, and usually they sound like they do, though sometimes there are ones that are nice as abstract problems. This varies from year to year. In 1994 there were some pretty problems (from the pure math point of view), though I heard this was less true in 1995.) We had a lot of freedom--to choose among the problems, to work alone or in a team. Five students worked on my particular problem. I had one close partner (who happened to be my roommate), while the others focused on different aspects.
Compared to an REU, the DSP felt a little like an office job--there was a huge "DSP Room" where we did all the research, with everybody having his own assigned desk and computer; the schedule was Monday to Friday, 9:00 to 5:00-ish, time cards, lunch breaks, and a daily commute. But only a little. The problem supporters were friendly and helpful. You work with new friends on exciting math problems that have real-life, top-secret applications, in a surprisingly informal atmosphere. That sure beats an office job--trust me, I've been through that too.
Since everything is top secret, they won't let you work at home, which means that evenings and weekends are pretty relaxed. Nevertheless, some of us found ourselves going in on weekends, particularly towards the end of the summer, so that we could think about our problems or see what the computers spat out over the weekend.
We spent the last week or two of the program writing up our results. Unlike an REU, you can't publish your results--all the DSP papers are carefully guarded at the Agency. It can be a little frustrating, after the summer is over, when you can't tell your friends (or grad schools) the cool math you did or even the title of your paper. (For one student, some of his results were recently "declassified" so that he can now submit them for publication "outside." Unfortunately, this doesn't happen often.)
Most of the DSP lived together at UMBC (University of Maryland, Baltimore County) for about $300 a month. We lived in suites of <= 4 people, with kitchen and (much needed) air conditioning. At UMBC there's the usual stuff--computer lab, game room (on an inconvenient summer schedule), library, piano rooms, basketball courts, and a volleyball sand pit. Besides these, not much else within walking distance. There are some recreational resources available on campus -- one of my roommates joined an Ultimate league. On weekends, we'd go for bowling, movies, and rollerskating. UMBC is 30 minutes away from DC and 15 minutes from Baltimore, so there are plenty of chances to visit both. You need a car to get to anywhere interesting (and to work), but if you're not bringing one chances are someone else will.
The application process involves tons of forms, getting yourself fingerprinted by HUPD, hopefully getting flown to the NSA for an interview, a polygraph test (to check whether you're a spy), a psychological exam (to check whether you're mentally stable), and having some friends and neighbors interviewed about you. I think it's worth it (actually it made me feel pretty important), although others thought differently. I guess it could be annoying when they just couldn't seem to get over the fact that your roommate's an international student from China (for example.)
One annoying thing about the DSP is that offers are sent out sometime in May, way past the response deadlines for REU's (and any other summer activity, for that matter). Usually the NSA math people have decided by then, and the delay is a matter of security stuff. So when you need to reply to REU people, just call/email the DSP mathematicians (whom you meet during your interview) and ask them "Am I in?"
Oh yeah, you get paid a lot (3 times as much as an REU).
Harvard participants in 1994: