A synchronous electric motor is an AC motor distinguished by a rotor spinning with coils passing magnets at the same rate as the alternating current and resulting magnetic field which drives it. Another way of saying this is that it has zero slip under usual operating conditions. Contrast this with an induction motor, which must slip in order to produce torque.
Sometimes a synchronous motor is used, not to drive a load, but to improve the power factor on the local grid it's connected to. It does this by providing reactive power to, or consuming reactive power from the grid. In this case the synchronous motor is called a Synchronous condenser.
Electrical power plants almost always use synchronous generators because it's very important to keep the frequency constant at which the generator is connected.
Low power applications include positioning machines, where high precision is required, and robot actuators.
Mains synchronous motors are used for electric clocks.