The best way to acquaint yourself with all the varieties of Chinese dumpling is to go out for dim sum. Dim sum, a light meal of many snacks served with tea, translates literally into "eat what your heart desires." Often, the restaurant will have carts circulating throughout the restaurant for you to choose what you'll eat.
Jiaozi are the most common Chinese dumpling, made out of rice flour dough wrapped around a cooked, spiced filling, then steamed. Har gao, or shrimp dumplings, are one of the most popular varieties.
Chive dumplings, or gao choi gao, also use the jiaozi wrapper, but are filled with a pungent pork and garlic-chive mixture.
Shu mai use a wheat-flour wrapper, and are pinched into an hourglass shape so the filling, usually either shrimp or pork, protrudes from the top.
American Chinese food has wholeheartedly embraced the wonton, which is, essentially, not unlike jiaozi, but with a thicker wrapper and a traditional pork filling.
Shanghai-style cuisine has two signature styles of dumpling. The first is the soup dumpling, sometimes called the juicy bun or xiao long bao, which consists of a pork or crabmeat filling, as well as a splash of soup inside the dumpling itself. The other is the potsticker, the steamed dumpling that's then sautéed in oil until one side is crisp and brown.