Lanolin is a greasy substance produced wool-bearing animals, and that's why it's also called wool fat, wool grease, or wool wax. It is a good emmolient and is widely used in the manufacture of lotions, creams, ointments, and other pharmaceutical and industrial products.

A lot of products are being marketed as hypoallergenic because they are "lanolin-free", suggesting that lanolin is a common cause of allergies. However, it seems that although there may be people who are truly allergic to lanolin, there's a scientific study that indicates that "at most, a weak contact allergen". According to this article from the BlueSky Medical,

At present, the generally accepted opinion among skin specialists is : If patients who are lanolin sensitive are unlikely to react to lanolin containing cosmetics, the risk to the general population must be extremely small. This view is reiterated by Kligman who wrote: "In a healthy population encountering lanolin in the normal way, the incidence of lanolin allergy is vanishingly small."

You'll need a patch test to determine if you really are allergic to lanolin. Once confirmed,you'll have to be a bit more vigilant in reading the ingredients of your moisturizers and other cosmetics. Avoiding exposure is the best way to prevent breakouts. Ask your allergist for other advice on coping with your skin sensitivities.