What is electromagnetic torque?

Current passed through a coil of wire (winding inside the motor) creates a magnetic field. In a properly designed motor, this field will be attracted to, or repelled from, another magnetic field also created in the motor. The attraction or repulsion is a force, and in a motor the force is directed to cause rotary motion -- hence, torque.

This is because the windings of the rotating part of the motor will create fields in opposition to the windings in the stationary part of the motor -- similar to the way north and south poles on a bar magnet attract each other while 2 norths or 2 souths repel each other.