If you're allergic, your body has an exaggerated response to some substance, like cat dander or pollen, that makes you sneeze or get itchy. Many people are allergic to dust.
If you have an allergy to something, your immune system goes crazy when you come in contact with it: if this happens to you, you're allergic. You might be allergic to certain plants, and spend the spring with itchy eyes, for example, or sneeze every time you pet your best friend's dog. Allergic comes from allergy, from the German Allergie, which is rooted in the Greek word allos, "other, different, or strange."