SYNCHRONOUS: means "turns at a speed proportional to applied frequency independent of load".
For this to occur, the magnetic field on the rotor must be powered separately from the stator magnetic field. As Hector pointed out - the rotor field may result from permanent (e.g. rare earth) magnets, or from an electromagnet.
If the rotor field is electrically created, the current has to come from somewhere. One way to do this is to use a separate DC supply and supply power through a rotating mechanical contact (brush and collector). The other way is to use a smaller machine driven off the main motor shaft (could be on the shaft, could be belt driven) to supply power to a rotating rectifier circuit to produce DC current for the main synchronous rotor field coils.
A specific value of rotor current corresponds to having the matched the required reactive load: more current means more reactive power. Reduce the current, and the machine must draw reactive power from the utility.
A Synchronous Induction Motor is a motor that will operate at Synchronous speeds: There is no slip rpm. A four pole standard induction motor, 50 hertz will operate at 1450-rpm or thereabouts under load. A synchronous induction motor 4-pole 50 hertz will operate at 1500-rpm as long as you don't exceed the rated load. It has a stator that is wound in the same fashion as a normal induction motor and it has a rotor that that has DC fields mounted on the rotor. The numbers of DC fields on the rotor correspond to the number of poles. The motor is started as an induction motor and as it reaches slip speed the rotor field is supplied with the rated DC Voltage. If done properly the rotor and stator will synchronize and the rotor will rotate at synchronous speed. The DC voltage for the rotor field comes from either a DC generator or a solid state static exciter.
The exciter provides the power to the electromagnets that form the poles on the rotor that in turn follow the rotating magnetic field from the system. The power factor of a system is a measure of the balance between the strength of the electromagnets in the rotor and the requirements of the rotating fields in the loads. Under excite and the system voltage drops resulting in reactive power into the alternator causing a negative power factor, over excite and the system voltage rises resulting in reactive power flowing out of the alternator causing a positive power factor; when the excitation is exactly matched to the reactive requirements of the system then the power factor is unity.