Allergy is due to your immune system reacting inappropriately to things it is coming into contact with.
Special cells in the body, called Antigen Presenting Cells, take chunks of protein (“antigen”) found in the tissues (esp gut, skin, lining of the mouth, nose and lung) and present them to T-lymphocytes, one of the main types of white cell found in the blood. The T-cell tries to recognize whether or not that protein is part of the body itself, or whether it comes from outside the body, in which case it shouldn’t be there! If the T-cell thinks that the antigen is from an infection, it will start the process of reacting to that infection with inflammation and antibody formation.
But sometimes the T-cell will start a slightly different process, with different kinds of inflammation and antibody. The inflammation in allergy is caused by Mast cells, another cell type that originates in the bone marrow but are found throughout the tissues of the body. Mast cells tend to sit inactive until they are triggered, at which point they release a whole mix of different chemicals into the surrounding tissue, often causing significant damage. One of these chemicals is Histamine, which causes dilation of blood vessels (producing redness and swelling) and also irritates nerve endings (producing itch). That’s why we use anti-histamines to treat or prevent allergic reactions.