If the motor contains a permanent magnet, it is not an induction motor. It is probably a permanent-magnet synchronous motor (PMSM). A brushless DC (BLDC) motor is essentially the same thing, but requires an electronic commutator or speed controller and operates from a DC supply.
Assuming it is a PMSM, the capacitor provides a phase shift to provide a second phase that is "split" from the single-phase supply. Look up "capacitor run single-phase motor" to find a detailed explanation. The second set of windings would be the auxiliary or second phase winding that is connected in series with the capacitor. Most such motors are induction motors. A motor with a permanent magnet rotor must be a PMSM. This is an unusual motor, but it may be a less expensive alternative for a ceiling fan.
As it is described, the motor can only run at a fixed speed. For variable speed, it would need to be configured as a BLDC motor with an electronic controller. BLDC motors are the preferred motor for computer cooling fans. Computer-fan motors have an electronic commutator built into them.