Re: capacitors
HID Ballast kits have Capacitors mostly to correct a poor Power Factor.
The Capacitor "Keeps" the VARs (Reactive Power) at the Ballast by "Storing It" in the Capacitor/Winding assemblidge (mostly in the Capacitor!).
HID Ballasts also use a Capacitor to perform additional Lamp operating tasks (High Reactance Ballast kits, for example).
On Induction Motors, the type of Motor is more of a factor than anything.
If the Motor is 1?, the Capacitor(or Capacitors if 2 or more used) is used in the Starting Circuit of a Split Phase Motor.
1? Induction Motors are not "Self Starting", like their Polyphase counterparts are, so they require some means of starting.
Some will have "Start/Run" Capacitors, which use a large value Capacitor for Starting, and a small value Capacitor for Running. These Motors keep the "Auxillary Winding" (The Split Phase Start Winding) in operation throughout the use of the Motor, whereas Split Phase Start type Motors only have the Aux. Winding connected during the initial starting of the AC Motor (centrifugal switch opens the aux. winding at apx. 80% full rotor speed).
PSC (Permanent Split Capacitor) type Motors - commonly used in Ceiling Fans, keep the Aux. Winding connected at all times - plus it is connected via a small value Capacitor.
Shaded Pole Motors do not require this stuff, as the embedded shading pole does the job!
The reason behind Polyphase Induction Motors being "Self Starting" and 1? not being Self Starting, is when the Rotor of the Motor is stationary, a Polyphase circuit will produce a Rotating Magnetic Field - which will make the Rotor begin to move.
A 1? circuit is Stationary, so in order to make the Rotor move, one "Side" of the Stationary Magnetic Field needs to be Reduced.
This is achieved by the Auxillary Winding (or shaded pole).
If a Polyphase Motor has Capacitors connected to it, they would be either for Power Factor Correction, or part of an AC line Filter.
Sorry to toss out such basic explanations, so feel free to ask for more info!
I am sure others have more to add.
BTW, take a look at items at the Technical Reference area of ECN, for some additional information.
Scott