Asynchronous motors are motors which run below the synchronous speed. The term "asynchronous motors" is usually used to refer to induction motors.
Motors which run at super synchronous speed are also asynchronous motors. The doubly fed induction motor is an example of a super synchronous motor.
By "asynchronous" we mean that the speed of the rotor is not equal to the synchronous speed of the rotating magnetic field of the stator. The difference between the speed of the rotor and the speed of the stator is called slip. In asynchronous motors, the slip is not equal to zero. In synchronous motors, the slip is zero.