The advantages of a three phase system over a single phase system are:-
Higher power/weight ratio of alternators. A three phase alternator is smaller and lighter that a single phase alternator of the same power output. Hence, it is also cheaper.
A three phase transmission system requires less copper or aluminium to transmit the same quantity of power of a specific distance than a single phase system.
Three phase motors are self-starting due to the rotating magnetic field induced by the three phases. On the other hand, a single phase motor is not self starting, it requires a capacitor and an auxilliary winding.
In Single phase systems, the instanteous power(power delivered at any instant) is not constant and is sinusoidal. This results in vibrations in single phase motors.
In a three phase power system, though, the instanteous power is always the same.
Three phase motors have better power factor compared to single phase motors.
Three phase supply can be rectified into dc supply with a lesser ripple factor.