I just got my fixed-gear folding bike from Bike Friday. It's based on their Pocket Rocket Pro frame, but instead of the normal gear system, it has a simple single gear. There are various kinds of "fixies". Some some have brakes and some have a gear on the other side which allows you to flip the back rim so you have two gears, although switching gears involves removing the back rim.
Because of the mechanics of a folding bike, the dropout where back rim mounts into the frame is slightly different from a normal fixie, but in principle it is the same. I was going to try to be badass and get it without any brakes (you use your legs to brake) but Stephen, my sales guy at Bike Friday, convinced me to get at least a front brake... that I could always remove it if I didn't need it.
After taking it around the block, I'm glad I have it. I don't think I have the legs to control myself down steep hills yet.
Fixies are getting more and more popular these days. They seem to have a slightly culty image, but I think more and more "normal" people are riding them and even major brands are starting to release fixies. The reason I got it is because it's supposed to be a good workout with the high cadence and the variety of weights you end up having to pedal. Also, because of the significantly fewer parts, it's more durable and less likely to break. Wikipedia has a pretty good article on fixed-gear bikes.
The bike folds so you can take it on the train or stick it in a cab and it also easily dismantles and fits in a airline checkinable suitcase.
This is my second Bike Friday. They're hand-made and high quality and the service and support has been good. I'd recommend them to anyone who wants a high quality bike that they can travel with.
Kudos to Sean for turning me on to fixies and to Markoff for turning me on to Bike Friday.
UPDATE: Here's a picture of it folded.