Gluten is a protein found in wheat, barley and rye. In persons with celiac disease (formerly celiac sprue, aka gluten sensitive enteropathy), eating gluten causes an inflammatory response in the bowel leading to abdominal pain, diarrhea, and malabsorption. The latter may cause anemia. The presenting symptoms can vary, but GI symptoms are the most common form of presentation. Celiac disease may affect up to 1% of the population. Simple blood tests are very good at screening for celiac disease. In celiac disease, the reaction to gluten is not an allergy, so allergy testing is not a part of the diagnostic workup.
Allergy to wheat should not be confused with gluten sensitivity. As with all food allergies, individuals with wheat allergy can have GI symptoms, but will lack the small bowel inflammation and malabsorption found in celiac disease. Most commonly, individuals with wheat allergy will present with a rash.
Lastly, there is nothing inherently bad about gluten. I have seen many patients who tried to remove gluten from their diets thinking it would help their allergy symptoms. There is no evidence to support a connection between eating gluten and allergies.
If you have questions about gluten sensitivity or celiac disease, your primary care doc is a good starting point.