Frostbite is the medical condition where localized damage is caused to skin

Skin

-Dermis:The dermis is the layer of skin beneath the epidermis that consists of connective tissue and cushions the body from stress and strain. The dermis is tightly connected to the epidermis by a basement membrane. It also harbors many Mechanoreceptors that provide the sense of touch and heat...

 and other tissues

Tissue (biology)

Tissue is a cellular organizational level intermediate between cells and a complete organism. A tissue is an ensemble of cells, not necessarily identical, but from the same origin, that together carry out a specific function. These are called tissues because of their identical functioning...

 due to extreme cold. Frostbite is most likely to happen in body parts farthest from the heart

Heart

The heart is a myogenic muscular organ found in all animals with a circulatory system , that is responsible for pumping blood throughout the blood vessels by repeated, rhythmic contractions...

 and those with large exposed areas. The initial stages of frostbite are sometimes called "frost nip".
There are several classifications for tissue damage caused by extreme cold including:

• Frostnip is a superficial cooling of tissues without cellular destruction.
• Chilblains are superficial ulcers of the skin that occur when a predisposed individual is repeatedly exposed to cold
• Frostbite involves tissue destruction.

At or below 0 °C (32 °F), blood vessels close to the skin start to constrict, and blood is shunted away from the extremities via the action of glomus bodies

Glomus body

A glomus body is a component of the dermis layer of the skin, involved in body temperature regulation. The glomus body consists of an arterio-venous shunt surrounded by a capsule of connective tissue. Glomus bodies are most numerous in the fingers and toes...

.