Lactase

Lactase is an enzyme that digests lactose, the main sugar in milk, into galactose and glucose.
Each molecule that your digestive system needs to work on before absorption is possible, which means almost all of them (the only exceptions being amino acids, small peptides, glucose and fructose, fatty acids and glycerol) has its own enzyme to break it down. Thus there are specific enzymes for breaking down fats, proteins, starches and sucrose, as well as lactose; lipase, protease, amylase, sucrase and lactase, respectively.
Lactase is an enzyme that stands out in the list, because deficiency in lactase is more common than the rest; a lot of people have a deficiency in lactase, and some people produce virtually none; people of African or East Asian descent commonly have a lot of problems digesting milk. This situation is genetic, and thought to be because people from these areas commonly didn't keep cattle until very recently, and therefore didn't drink milk and had no need for lactase.
This situation arises because no other animal, including our ancestors, had a need for lactase beyond the age of about 2, because no other adult animal drinks milk. Babies and toddlers, even of the ethnic groups mentioned above, rarely have problems digesting milk. Lactase deficiency becomes more likely as you get older, and especially if you are of East Asian or African descent, even partially.
Lactase deficiency is also sometimes caused by various diseases of the digestive system, particularly of the small intestine, such as Crohn's disease, coeliac disease and (temporarily, usually) gastroenteritis. Chemotherapy and large amounts of antibiotics over long periods of time can also cause lactase deficiency. The latter is because the normal intestinal flora do some of the work of digesting lactose.
Lactase supplements are useful if you have problems digesting lactase, for any reason.