The component of jewelry that people are most often allergic to is nickel. Nickel allergy is actually a very common sort of allergy, which is a shame because most yellow gold and white gold products are alloyed with nickel. The higher the karat, the less nickel is used, but the more malleable the gold is. Having an allergic reaction to nickel is called dermatitis. There are other types of gold mixtures that do not have nickel like green gold, gray-white gold, and pink gold so take heart in your other options! There are also other non-gold choices like platinum, palladium, cobalt, etc.
There are preventative measures for dealing with dermatitis. For instance, wearing jewelry while it is smoldering hot outside is a terrible idea. The sweat will often wear away the nickel and form nickel salts which then reacts with the skin and causes an allergic reaction. Keeping the skin clean and dry will help prevent irritation, or simply removing the jewelry when outside for long periods of time.