People get frostbite when their skin is exposed to extremely cold temperatures for long enough that their tissue is damaged. High altitude hikers in snowy conditions are vulnerable to frostbite.
If your friend scolds you for not wearing your gloves on a bitterly cold day, she's not just giving you a hard time — she might be worried that you'll get frostbite on your fingers. Severe frostbite results in dead tissue and skin, and even the amputation of fingers or toes. Frostbitten came before frostbite, from the idea that your painfully cold fingers have been bitten by frost.