Starting capacitors are typically wired in series with the winding, at least in my memory. This causes a phase shift in the current on that winding. The run winding is active at the same time, and this "fools" the motor into thinking that it has more than a single phase coming in. The additional phase helps drag the rotor to get it started (and in the correct direction). Once the motor is turning, the additional phase is not needed.

Three-phase motors need no starting windings, but like single-phase motors, they often do have run capacitors. These are wired in parallel with the windings, and their sole purpose is to help correct the power factor.