A fixed gear bicycle, also known as a fixie, is a single speed bicycle without a freewheel. The cog is 'fixed' to the hub, usually secured by a reverse-thread lockring.
Fixed gear bicycles cannot coast. Whenever the rear wheel is turning, the pedals turn in the same direction. By resisting the forward motion of the pedals, a rider is able to slow the bike to a stop, without the aid of a brake. They can also be ridden backwards.
Track bikes are a specific type of fixed gear bicycle, designed only for riding in a velodrome, but many fixed gear bikes are suitable for on-road or off-road use. Because it is possible to stop the bike by resisting the turning pedals, brakes are not strictly necessary on fixed-gear bikes, but many cyclists retain at least a front brake for safety in traffic (and in case of snapped chain) and to reduce fatigue.
Fixed gear bicycles are especially popular among bicycle messengers, who prefer them for their light weight, simplicity and maneuverability when used by a skilled rider.
Conversion
Fixed gear bicycles (like all bicycles without derailleur) require a mechanism to allow for the adjustment of chain tension. This is important to keep in mind when converting from a freewheel to a fixed hub. Most bicycles with horizontal dropouts can be easily converted by changing out the hub. The tension can be adjusted by moving the wheel forward or backward in the dropouts. Bicycles with vertical dropouts can also be converted with some additional hardware. Possibilities include:
An eccentric hub or bottom bracket allows the off center axle or bottom bracket spindle to pivot and changing the chain tension. See ENO Fixed/Free hubs listed in the external links.
A "Ghost" or "floating" chainring. an additional chainring placed in the drive train between the driving chainring and cog.
A "Magic gear". With some math and a lot of luck you can calculate a gearing ratio to fit a taut chain between the rear dropout and bottom bracket.