The electron moves back and forth, probably an astonishingly short distance, 60 times a second.
The question I think you're asking is, (a) what makes it move? and (b) how does that movement do any useful work?
(a) It moves because an electric field pushes it, or pulls it, alternately. That electric field comes from upstream electrons crowding towards it, each in turn pushed by the electric field, all the way back to the generator. Or being pulled away from it, in the other half cycle.
Line up a row of pennies with 1mm between them, and push the left hand one 10mm towards the others. You'll see that though each penny moves a few mm, the wavefront of increased money density has moved 100mm or more. That rate is limited by the speed of sound in pennies. (Yes it's crude, pennies only repel each other by physical contact, while identical charges repel each other by an electric field)
So, though each electron moves at a relatively slow "drift velocity" the energy (transmitted by electric field) propagates at the speed of light (not in vacuo, but slower thanks to materials properties) - typically 0.7*C.
(b) It loses energy in two ways. First, it bumps into the atoms in the wire, transmitting kinetic energy to them, making them vibrate. In your fan, this is wasted energy; in a stove or lampbulb, it heats the wire - to about 300K in the latter case, emitting light.
Secondly, as a moving charge, it generates a magnetic field. If there are nearby conductors, that magnetic field induces current in those conductors, which generate their own magnetic field, and generates a force between them. The strength of that magnetic field retards the electron (this is the basis of inductance). If the second conductors are free to move in the direction of that force, they will, turning the fan blades. Force * distance = energy - this energy is drawn (via the magnetic field) from the electron's motion. (Of course, rather than generating magnetic field in a movable set of conductors, some motors use movable permanent magnets instead. Or fixed magnets and movable conductors, etc)
If you can see the link between electron flow and mechanical motion, you also need to understand that this process is perfectly reversible, and ni fact the generator produces the "push" aka electro-motive force (EMF), aka voltage, by the same process in reverse.