A Fixed Gear bicycle is usually a single-speed bicycle (only one gear) without a freewheel. Also quite often seen in Critical Mass rides.
If you pedal backwards on a Fixed-gear you will go backwards. On a single-speed this does not have to be. They also have only one speed, but you can't go into reverse.
Fixed gear bikes originally corresponded with Track Bikes, or bikes that were used in races on a velodrome. But once bike messengers in the late 70's early 80's found more and more about them, they adopted the bikes because of their simplicity and durability.
Anyone who rides a fixed gear understands the concept of "being one with the bike".