An allergy is basically a reaction of the immune system gone haywire. It believes that the food proteins are harmful to the body and mounts an attack to fight them off. This leads to an allergic reaction causing a multitude of unpleasant and sometimes even dangerous symptoms.
The misguided reaction of the immune system to a certain food protein and the symptoms caused by this are called food allergic reaction.
Whereas food allergies are always caused by proteins, food intolerance can be triggered by various different components contained in the food eaten.
Very often food intolerances or other non-allergic food reactions are confused with food allergies which leads to the mistaken belief that food allergies are more common than they actually are.