Celiac disease is an autoimmune disorder of the small intestine that occurs in genetically predisposed people of all ages from middle infancy on up. Symptoms include chronic diarrhœa, failure to thrive (in children), and fatigue, but these may be absent, and symptoms in all other organ systems have been described. A growing portion of diagnoses are being made in asymptomatic persons as a result of increased screening.

Coeliac disease is caused by a reaction to gliadin, a gluten protein found in wheat (and similar proteins of the tribe Triticeae, which includes other cultivars such as barley and rye). Upon exposure to gliadin, the enzyme tissue transglutaminase modifies the protein, and the immune system cross-reacts with the small-bowel tissue, causing an inflammatory reaction. That leads to flattening of the lining of the small intestine (called villous atrophy). This interferes with the absorption of nutrients, because the intestinal villi are responsible for absorption. The only effective treatment is a lifelong gluten-free diet. While the disease is caused by a reaction to wheat proteins, it is not the same as wheat allergy.

Resources:
Celiac.Com
Celiac Disease Foundation
General Information on Celiac
Wikipedia Article
Medical Info on Celiac
Information on Glutten-Free Products