A "fixie" bike is a nickname for a fixed-gear bicycle, which is a single-speed bicycle that operates without a freewheel mechanism. A freewheel allows the wheels to spin without moving the pedals. This means that a fixed-gear bike in motion continuously moves its pedals as well.
The advantages of a fixed-gear bike, like all single-speed bikes, are in the simplicity, light weight and lack of moving parts to foul up. Fixed-gear bikes do not have gear-shifting mechanisms or sets of sprockets for different speeds. The pedals can be used for braking, so some fixed-gear bikes don't have handbrakes, although this is not a safe or advisable method of operating a bike.
The disadvantage of fixies is that they are challenging to operate on hilly terrain. The single gear is generally not well suited for steep inclines.