The structure that extends from the base of the brain through a large opening in the skull and proceeds inferiorly in the vertebral canal is the cord.

The structure you are referring to is the spinal cord. It is an essential part of the central nervous system and extends from the base of the brain (specifically from the medulla oblongata) through the foramen magnum of the skull. The spinal cord continues downward within the vertebral canal, which is formed by the vertebrae of the spine.

The spinal cord is responsible for transmitting nerve signals between the brain and the rest of the body, facilitating motor control and sensory perception. It also plays a critical role in reflex actions. The protective encasing of the spinal cord is provided by the vertebrae and the meninges, three layers of membranes that surround the brain and spinal cord. Understanding the anatomy and function of the spinal cord is fundamental in the fields of neuroscience, medicine, and anatomy.