An allergic reaction is what the body does when exposed to an allergen. From the outside, this is seen as the allergy symptoms. Internally, this involves the body’s immune response the the allergen which triggers the symptoms.
Strictly speaking, allergies are a type I hypersensitivity, involving an immediate onset allergic reaction (atopic, or anaphylactic). The broader definition used on this site includes all five classifications of hypersensitivity.
Internally, a variety of complicated mechanisms can come in to play, involving any one of several immune system responses. The comlexity of the immune system cause unpredicatable sets of symptoms from one individual to the next.