Chris King rear hub with sprocket.
This means you must pedal going down hills. I am not sure what the mechanical differences of a fixed hub and "regular" hub are but I would be interested in learning.
Some Fixies have brakes, and some do not. They are stopped by putting backward pressure on the pedals, which takes a bit of practice.
Lots of fixies sport colorful tires, rims, pedals,chains, hubs, seats and paint jobs. If you are building up a new bike you are only limitted by your imagination.
Converting an existing bike into a fixed gear requires a frame with a horizontal drop out as seen in the picture below.
A lot of older 10 speed bikes have horizontal drop outs which make them ideal candidates for a fixed gear conversion. Because of the horizontal drop out and the increased amount of torque caused by a fixed gear the axle is secured to the frame with a nut, rather than a quick release skewer.
Fat tires are fine if your frame allows
The type of tires, handle bars, pedal, etc. is up to the rider's preferences. The choice of chainring and sprocket will depend on the type of terrain ridden and the riders fitness.
Why ride a fixie? For some it is a cultural thing, for others its about fitness, efficiency, and the connection with the bike. For others its just fun. Do you ride a fixie, if so what do you like about them?