A person who has allergies has a compromised immune system because the body's immune system:

The body's immune system thinks what he/she is eating or inhaling is a pathogen. The immune system sends white blood cells to attack. This triggers an allergic reaction.

For example: I am allergic to asparagus (I'm not joking, I really am) and walnuts. Anytime I eat one of those, my body's immune system thinks it's a pathogen for some reason and attacks and triggers an allergic reaction in me. This makes my breathing hard and makes it really itchy. Good thing it's not that severe so I can eat some if I want, but sometimes allergic reactions could cause hospitalization or death.