Peduncle

n.
  1. (Bot.) The stem or stalk that supports the flower or fruit of a plant, or a cluster of flowers or fruits.
  2. (Zoöl.) A sort of stem by which certain shells and barnacles are attached to other objects. See Illust. of Barnacle.
  3. (Anat.) A band of nervous or fibrous matter connecting different parts of the brain; as, the peduncles of the cerebellum; the peduncles of the pineal gland.

The ultimate divisions or branches of a peduncle are called pedicels. In the case of a solitary flower, the stalk would be called a peduncle if the flower is large, and a pedicel if it is small or delicate.