An "allergy" is your body's negative reaction to some kind of substance, like pollen, dust, mold, or certain kinds of food. When you have contact with the substance that you are allergic to, you start to feel symptoms like sneezing, headaches, swelling, itchiness, and so on.
If you say someone "has allergies", it usually means that they're allergic to pollen. If the allergy is something else, you say that they "have a ___ allergy" or they "are allergic to ___". For example:
Martin has a peanut allergy.
I'm allergic to cats.
Are you allergic to shellfish?