Next to salt, pepper, especially ground black peppercorn, is probably the most often used ingredient in cooking worldwide.
There are actually four types of peppercorn used in modern dishes: black, white, green, and pink. The black peppercorn is by far the most used of the four, and has a long history as a gourmet spice. In fact, it was one of the dominant trade items between East and West five hundred years ago, along with tea and other spices, and was valuable enough that several expeditions like that of Columbus were tasked with finding a quicker trade route for such things.
Of the four peppercorns, black is also the strongest, with a very distinctive spicy, and slightly sweet flavour. The white and green varietals have more sweetness and less fire, but are often used not for the flavour difference, but for the presentation. White peppercorn, for example, is often used in white sauces like alfredo sauce, to give a pepper taste without having little black flecks in the sauce.
Black, white, and green peppercorns are the dried berries of the pepper plant (Piper nigrum, as Webster handily points out), which looks a fair bit like a grape vine (the berries hang in grape-like clusters) and is generally grown in India and other Eastern countries. Whereas the black peppercorns are made with the almost-ripe berry, the white are made with the ripened and peeled berry, and the green with the very unripe berry. The white and black varietals are dried, whereas the softer green ones are usually in a brine and consequently have a shorter shelf life. Pink peppercorns are actually from an entirely different plant, a type of rose, I believe, and are generally also used for presentation.
Black peppercorn is said to aid digestion but I've never seen a first hand source to back it up. If you're in the market for pepper, you want to aim for Tellicherry peppercorns, or Lampong, but unfortunately, most brands don't indicate the type used.
And of course, you don't buy pre-ground pepper unless there is a gun to your head. The peppers, once ground, tend to lose a lot of their flavour pretty quickly (quickly being days, not minutes, don't worry about rushing when cooking) because like most spices, the flavour is in the plant oils which are often volatile at room temperature (volatile as in they evaporate, not explode). A pepper grinder will cost you three dollars and is always worth the price.