A synchronous AC motor is a special type in which the output rotor speed is directly aligned - ’in sync’, as it were - with the rate of the alternating current being supplied to it. Whereas this would tend to induce no current under a standard AC induction or asynchronous motor construction, the synchronous AC motor typically has additional components fitted, known as slip rings, which allow for current transmission between the motor’s rotating and fixed parts.
Slip rings are electromechanical devices that allow electrical signals to be transmitted through electromechanical systems constructed from components that need to rotate while generating power. In the case of a synchronous AC motor, the slip rings are what generate the necessary magnetic field around the rotor, and thus allow it to turn at the same rate as the alternating current being supplied without breaking the flow of current.
In summary then, AC synchronous motors are so-called because the speed of the rotor is directly proportional to the rotating magnetic field of the stator. They tend to be the preferred option wherever a higher degree of accuracy is an important factor in the application, such as in clocks and other timing devices.