When a magnet rotates inside a generator with three coils positiond at 120 degrees from each other, then, by Faraday's law, we generate three-phase alternating currents (3-ph AC). If we have a motor with three coiles at 120 degree and input our 3-f AC, then, by Ampere's law, there will be a rotating magnetic field and cause a induced current in the rotor and creates a torque to rotate the rotor.
One can think of three phase AC circuits as having three independent AC power sources, which have the same amplitude, same frequency but their phases are 2*π/3 apart.
Or, we can think of an AC voltage as a rotating vector (its x-componet, vx = v cos(2*π*f*t), is what we detect), then the 3-ph AC source has three rotating vectors, at 120 degrees from each other, all rotate at the same frequency (e.g., 60 Hz in USA). Their sum is always zero as we can see from their x-components in the graph.
A 3-phase motor can be very powerful(compared to one-phase motors) and easy to maintain or repaire (compared with DC motors). How do we control the rotational direction ? How do we protect the motor, e.g., from two-phase running? The principles are in
Electricity, Magnetism and Electronics.