Black peppercorns are the nearly-ripe berries of Piper nigram. Tellicherry is generally considered to be the best variety of black pepper and comes from the Malabar Coast of India. Lampong and Sarawak are also popular varieties.
White peppercorns are the seeds of the fully ripe berry. They show up in light-colored sauces, purely for aesthetic reasons, and in Scandinavian cooking.
Pink peppercorns are the dried berries of the Baies rose and technically not pepper at all. They’re less widely used but can be found in vinaigrettes and light sauces for fish.