Electric motor - Function

First of all, all motors function based on interacting magnetic fields. These fields are created by the supply currents that power electromagnets.

Now, we have 2 major types of motors, the DC and the ac induction and ac synchronous motors.

I will be drawing simplified analogies to explain the working.
The DC motor can be pictured as a magnet which is fixed at its middle but can freely rotate about this centre, much like a compass needle. In the case of a dc motor, this represents the rotating shaft, the rotor. This rotor is placed in a magnetic field (stator field) and the rotor is energised in such a way that it's magnetic field is at an angle to the stator field. This causes the rotor to try and align itself to the stator field. Much like what happens if you being a compass near a magnet and it alighns itself to the magnet's field. This tendency for the rotor to align itself to the stator field causes the creation of rotational forces( torque). But unlike the compass, we see that the rotor continues to rotate. This is because the rotor is designed in such a way that when it rotates by a certain angle, the connections that feed its magnetic field are modified(using split rings)and his leads to the rotor's magnetic field taking a new position with respect to the stator field. This continuing misalignment between rotor and stator field leads to a continued effort by the rotor to realighn itself to the stator field, which actually never happens. Note here that the stator field is fixed and the rotor field is allowed to exist on a specific range of angles with respect to the stator.

The ac motors work based on generatong a rotating magnetic field. The wiring of the stator coils is done in such a way as to have the coils of each phase of the ac supply at an equal angle from the other. When one adds the magnetic fields vectorially, the resultant field is observed to have a fixed magnitude but a rotating nature, rotating at the supply frequency times a constant( affected by the number of poles). Now that we have a rotating magnetic field we can look at the ac motors.

The synchronous ac motor is basically like taking the compass and rotating a bar magnet around it to get it to move. So the rotor of the synchronous machine is nothing but a electric bar magnet, which in turn interacts with the rotating stator field, maintain the same speed as the stator field.(hence synchronous)

Finally, we have the ac induction motor. The ac induction motor has a rotor which is concentric with a squirrel cage like structure. When this squirrel cage is placed in a rotating magnetic field, the magnetic field lines of the stator cut the rotor' squirrel cage structure leading to currents being induced in the rotor. These currents generate a magnetic field as well and these fields interact with the rotating stator field much like how the DC motor works, and lead to the rotation of the rotor. As the rotor pocks up speed, the relative speed between the rotor and stator fields reduce and fewer stator lines cut the rotot for a unit time. This leads to weaker rotor filed a at higher speeds, and if the rotor were to reach the stator field's speed, the rotor would not have any currents included in it because it sees a stationary field and so no torque would be produced. Hence the rotor would slow down to a balance point where the rotor speed is just slightly lesser than the stator's rotating field's speed. This continued effort by the rotor to keep up with the stator is what makes the motor to rotate as long as it is energised and well maintained.

Most do the details I have provided is a compressed understanding of volumes of notes and equations. But the major terms I've alluded would be enough to get your exploration started.