All three are produced from the berry of the same tree. Their distinguishing characteristics derive from variations in harvesting time and processing methods.
The most common of the three, the wrinkle-skinned black peppercorn, is picked slightly immature and then dried whole. The relatively smooth-surfaced white peppercorn is picked at full maturity. After it is soaked to facilitate the removal of its skin, it is dried skinless. Green peppercorns are picked immature and then preserved, skin and all, by pickling.
Some cooks prefer to use white peppercorns in pale preparations such as white sauces and cream-based sauces. If black peppercorns were used, the black specks of the ground dried skin might detract from the food's visual appeal. Other cooks prefer to take advantage of the wonderful intense flavor and fragrance that the skin of the dried pepper provides. Green peppercorns, the mildest and classiest of the three, are often cooked and served whole in dishes like braised duck.
Pink peppercorns, by the way, are not true peppercorns - they come from a different plant. Although the dried pink peppercorn berry resembles the true item in size and shape, it has a different color, taste and aroma.