Coeliac disease is caused by a reaction to gliadin, a prolamin (gluten protein) found in wheat, and similar proteins found in the crops of the tribe Triticeae (which includes other common grains such as barley and rye).
Upon exposure to gliadin, and specifically to three peptides found in prolamins, 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 a truncating of the villi lining the small intestine (called villous atrophy). This interferes with the absorption of nutrients because the intestinal villi are responsible for absorption. The only known 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.