Paint will not adhere to bare wood unless it is primed first. Generally, you do not need to prime painted wood to promote adhesion; however, there are instances where the opposite is true. You need to properly prepare painted wood for a fresh finish, or peeling may result.
Water-based Finishes
If you are applying a water-based latex or acrylic finish on top of another water-based finish, you do not need to apply primer, first. However, you may still need to condition the wood for better adhesion to prevent eventual finish failure. If the wood is finished with a glossy latex paint, it requires abrasion prior to application. Sand the existing finish until it feels rough, or the new paint may not stick.
Oil-based Finishes
If you plan to apply an oil-based paint on top of an existing oil-based finish, no primer is necessary. As long as the new and existing finishes have the same base, they will adequately bond. If the existing oil-based finish has a high sheen, abrade it just as you would a glossy water-based finish.
Incompatible Paints
If the new paint has a different base from the existing finish, adhesion will prove impossible without the help of a bonding primer. If you apply a water-based paint over an existing oil-based finish, peeling will result. The same holds for applying oil-based paint over an old water-based finish. To make the two paints bond, apply a shellac bonding primer first. Once the existing painted finish is primed with shellac, it will accept any type of paint.
Considerations
If you plan to apply a light color of paint directly on top of a dark color, you can promote better coverage by adding a white base primer. This is also true if you plan to apply a dark color of paint on top of a very light finish. In the second case, you should choose a tinted primer. Although it's not necessary to promote adhesion, this technique can reduce the amount of coats needed to generate a uniform finish.