There are a great many dishes known around the world as dumplings. Some are sweet desserts, some are balls of steamed dough and others are thin wrappers of dough containing any number of fillings. There are only two things they all have in common. They all involve some form of dough, and they are all comfort food of the highest order in their respective cultures.
Types of Dumplings
There are two main types of dumplings: those made from a solid ball of dough and those made with wrappers and filling. Within those two broad categories, there are a range of foods with different cooking methods. The dough type of dumplings are usually light and leavened in America, but are often made dense and heavy in other places. German spaetzle are especially small dumplings, almost noodles, made from a stiff batter. The filled type of dumplings include Polish pierogi, Japanese gyoza and the hundreds of delicate Chinese dumplings. Even ravioli can be considered a dumpling.
Boiling is one of the primary cooking methods for dumplings. Heavy dumplings such as the African fufu, or the German potato dumplings, are commonly cooked in boiling water until the centers are no longer doughy. Italian gnocchi are also boiled in water or broth, and so are sturdy filled dumplings such as pierogi. The rule of thumb is that dumplings to be boiled must be sturdy enough to hold together in the rolling, bubbling pot of water. Those that aren't should be steamed instead.
Steaming is a better technique for most dumplings than boiling, simply because they are less likely to fall apart during cooking. Most Asian dumplings are steamed in a lightly-oiled bamboo steamer until they are fully cooked, then removed and served in broth or with various dipping sauces. Most cooks think of their light American-style dumplings as being boiled, but this is not the case. Although they're normally prepared on top of a soup or stew, most of the cooking is done by steam trapped in the pot by a tight-fitting lid.
Some dumplings are prepared at least in part by frying, giving them a golden crust and more interesting flavor. A classic example are "potstickers," which are browned on the bottom in a hot skillet, then steamed until done. Italian gnocchi and German spaetzle are initially prepared by boiling, but are often finished in a skillet. In the case of spaetzle, they'll be steamed under a cover for a minute or two until puffed, then finished uncovered with a pat of butter and some breadcrumbs for a garnish and textural contrast.