Final answer:
The portion of a hair that is embedded in the skin is called the "hair follicle," with the living part below the skin's surface being the "hair root."
Step-by-step explanation:
The name for the portion of hair embedded in the skin is the hair follicle. More specifically, within the follicle, the hair root is the portion of the hair that is below the surface of the skin and ends deep in the dermis at the hair bulb, which surrounds the hair papilla. The hair matrix, a layer of mitotically active basal cells, is located within the hair bulb. The part of the hair that emerges above the skin surface is known as the hair shaft, which is composed of dead, keratinized cells and has no biochemical activity making it essentially dead.