White meat commonly refers to two separate, though closely related, concepts.
Informally, 'white meat' refers to the lighter cuts of meat, especially when talking about poultry. Chicken breast, for example, is much lighter than the thighs and drumsticks ('dark meat'). This may also be termed 'light meat', in order to distinguish it from the second meaning.
More formally, white meat may refer to any poultry or fish, and sometimes pork and veal, all of which have lighter color than beef (and most other meat animals), which are referred to as 'red meat'. The color in red meat comes from the protein myoglobin, which is found in slow-twitch muscles (important for large, heavy grazers) and is red in color when raw, turning to brown when cooked.
Myoglobin is the same protein that causes the difference in color between the chicken thigh and breast, but poultry and fish have more fast-twitch muscles, which have very low levels of myoglobin, and thus lighter meat. Pork is actually midway between red and white meat, having moderate myoglobin levels. The marketing slogan "Pork. The Other White Meat" is just that -- marketing. Nutritionists generally consider all meat from mammals to be red meat, although some sources will consider rabbit to be white meat.
The main health benefit to white meat is that it has lower fat content, although there are other minor health risks associated with red meat; on the other hand, red meat has more vitamins and minerals than white meat.
Somewhat more obscurely, until the 1800s white meat (or whitemeat) referred to foods prepared from milk -- what we today call dairy products. This category also sometimes included eggs.
Wikipedia: White Meat
Exploratorium: The science of Cooking: What Gives Meat its Color
The Oxford English Dictionary.