Synchronous motors and asynchronous (often called induction) motors are two kinds of AC motors. The more common of these two is the induction type due to its lower cost and simplicity.
Induction motors have no brushes to wear out. This is due to a design in which current is induced into the rotor (the part that spins) from the electromagnets on the stator (the stationary part). Induction motors are also known as asynchronous motors because the speed of spin is not synchronized to the frequency of the power as is the case with synchronous motors. In an induction motor the stator is typically made of thin steel laminations with coils of wire on them, which are then placed inside a cast iron frame. The coils are wrapped in such a way that it causes a rotating magnetic field when AC power is applied. This rotation is due to the inherent properties of ac power and its frequency, switching between positive and negative.
The rotor is made out of conductive bars that are shorted together on the ends but have no electrical connection otherwise. The bars are assembled at a point where they are slightly twisted and resemble a squirrel cage that an animal would run in. When this rotor assembly is placed inside the rotating magnetic field from the stator a current is induced in the bars such that it creates a magnetic field in the rotor. These two magnetic fields produce a force which causes the rotor to spin. Induction motors run at a speed slightly slower than would be expected based on the frequency of the power. This difference in speed is called slip and is related to how the field induces power to drive the rotor. While the concept of an induction motor can be a little complex in practice the design is very simple and efficient to manufacturer. Induction motors are used in many household devices such as heating systems and pumps, since having few wear parts they have long service lives.
Synchronous motors are synchronized to the frequency of the AC power, where the speed is always some multiple of the frequency. In small applications this can be useful for driving something like a clock. Synchronous motors have a stator and a rotor as well, but both the stator and rotor have wire coils making up electromagnets. The stator is similar to that of an induction motor, with coils of wire around thin steel laminated metal. The main difference is in the rotor which contains coils of wire and slip rings that supply power to the rotating coils.
Since the field is rotating in the stator, it must be fixed in the rotor so the motor will spin. To achieve this non rotating field DC power is supplied to the coils on the rotor. In a synchronous motor the motor will either spin at the correct speed or will not spin at all. Variable speed controllers are often used to adjust the frequency driving the motor. Sometimes the motors have feedback as well, so the controller can more accurately adjust the speed. Synchronous motors can provide a lot of torque but if the driving force drops too low, it will lose sync and stop spinning. Due to the added complexity, synchronous motors tend to be more expensive, and they often require control circuitry to drive them further increasing the cost. Synchronous motors are more efficient than induction motors and have a higher power density. An equivalent-torque synchronous motor will generally be smaller than an induction motor but that can also cause an issue with heat if not dissipated correctly. Synchronous motors are generally used for driving more precise loads such as a robot or for constant speed applications like industrial equipment.
Both types of motors have their benefits and detractors but ultimately the application should determine the best motor for the job.