That Wiki is quite good but it does not explain how the single phase motor starts and it leaves out the Split Phase motor which is the style I was referring to.
The 3 phase explanation describes how the magnet field 'rotates' because the field coils are energized in sequence; single phase,of course, just reverses the magnetic field but doesn't rotate it. In the capacitor start, or run, single phase motor there is a second coil that has the capacitor in its circuit and this means the magnetic field in this coil either precedes or follows the field in the main coils; this induces a rotation in the magnetic field so the motor starts. In the Split Phase motor the second coil does not use a capacitor but the metal pole of this field has a copper strip across it that influences the way the magnetic field behaves and causes it precede or follow the main field again creating rotation and starting the motor. Once the motor is up to speed it is possible to take the starting coil out of the circuit and on large motors this is done by the centrifugal switch. On a motor that leaves the starting coil in operation normally the starting torque is low, the starting coil does not draw much current, so that this coil does not waste too much energy when it is not needed. On larger motors which need good starting torque the starting coil is designed to draw a large current and if it is left in the circuit once the motor is up to speed it draws too much current and the breaker will kick out or the motor overheat sensor will trip.
If you haven't already done it go and take a few courses in Physics and basic electrical theory they can be quite fascinating and very useful in the long run.