Electric fan not working?

There is great deal of variety in the way speed control work for fans. I am going to ume that what you have is a cheap-o bottom of the line fan. In this type of fan the speed control is not continuous, but consists of just a few (typically three) speed settings plus an "off" position. In this set-up, the speed is controlled by the number of stator windings energized. The more windings energized, the slower the speed, but with no windings energized the speed is zero. The typical failure mode for this type of fan is an insulation failure in a stator winding, this leads to overcurrent, vibration, and failure in other windings. The only way to fix this type of failure is to rewind or replace the stator. It is usually cheaper to replace the entire motor and this eliminates the possibility of rotor damage caused by phase slippage when the stator failed.