Anorexia is a serious medical disorder in which someone has no appetite, and so doesn't eat. If someone you know becomes so thin that it alarms you, they might have anorexia.
Anorexia means "no appetite" in Greek. The full name of the medical disorder is anorexia nervosa, a very serious problem that causes people to starve themselves as the result of an emotional disturbance rather than a problem with food or digestion, or simply not being hungry.