According to this article by veterinarians Dr Tiffany or Dr Lana from the Cayman Animal Hospital, there are five known types of allergies in cats: contact, flea, food, bacterial, and inhalant.

Except for fleas, I think these are also the main allergens in humans. Even the symptoms sound familiar:

The most common is itching of the skin, either localized (one area) or generalized (all over the cat). Another manifestation involves the respiratory system and may result in coughing, sneezing, and/or wheezing. Sometimes, there may be an associated nasal or ocular (eye) discharge. The third manifestation involves the digestive system, resulting in vomiting or diarrhea.

If you yourself have allergies, it seems it'd be a breeze spotting the signs on your cats. The most troublesome, though, is that which we humans don't have (or at least not normally): fleas. Although non-allergic cats will suffer only mild itching and irritation due to flea bites, those with allergies will be prone to secondary infections because of the intense itchiness. This may subsequently lead to scabs, sores and hair loss.

If your cat (or dog) has fleas and is showing great irritation, consult your vet for proper treatment. A topically applied medicine may be needed.