NCBC Fixed Gear

A bike like the Wright Brother's rode, designed, built, loved and sold, back in the days before Lycra
No derailleur, no shifting, no freewheeling - just a solid connection between the chain and the rear wheel.

Racers ride them for winter training.
Sheldon "Coasting Is Bad For You" Brown (Alpha–wrench and many a cyclist's Guru) prescribes fixed-gear for "fun, fitness, form, feel & 'ficciency!".
Lacking fragile derailleurs, fixed gears make great travel bikes and are gaining popularity in off-road applications - see 63xc.com – the Offroad Fixed Gear Site
It's a great way to give a whole new life to that old steel–tube ten–speed that you never ride anymore.
(Sheldon's site is the mother lode of conversion information)
They've got a certain rebel-youth attraction (What cyclist isn't a youth at heart?) – Messenger turned author Travis Hugh Cully fawns over fixies in his book "The Immortal Class: Bike Messengers and the Cult of Human Power".
Striped of cables and extra ecoutraments, the bicycle takes gains a beautiful simplicity.
Then, there's the "Retro-Grouch" snob appeal...

Competition Track cycling is done on derailleurless track bikes, some of which can be quite extreme (also.)
"Stayers" are easily recognized by the reverse-raked fork which allows for close pacing behind a motorized pace vehicle.
They're used in setting cycling land speed records – See Dr. Abbott.
Often at the other end of the spectrum, there's a huge collection of Fixie Photos in the Gallery – mostly road bike conversions.
Here's some submissions from the Triangle:

• #98 – Lynn Forrest's Rochet
• #99 – Lynn Forrest's Cinelli
• #268 – Linwood Hines' LandShark
• #286 – Shawn McCarthy's Raleigh
• #595 – John Rider's Centurian
• #618 – Adrian Hands' KHS

NPR's Noah Adams ("All Things Considered") interviews Wes Williams, owner of the Crested Butte Bicycle Company, regarding offroad Scorchers (or here).