relation between phase and pole
No, that is not true. In a three phase induction motor, three sets of windings are used for each pair of poles. The magnetic fields of the three phases are summed together because of the placement and distribution of the windings. The phase displacement of the currents in each phase causes the summation to change during the course of each sinusoidal cycle such that the magnetic poles that are formed rotate around the motor.

Most single phase motors are designed as two-phase motors. The applied power is single phase, but the power applied to one of the two sets of phase windings is connected through a capacitor to create a phase displacement. Single-phase motors are a bit more difficult to explain than three-phase motors. I will not attempt to give a more complete explanation right now.

For a complete explanation of both single-phase and three-phase motors, it is best to consult a text book or detailed course notes which include all of the necessary diagrams and mathematics.