Final answer:
The neural crest cells are responsible for giving rise to the peripheral nervous system and certain facial structures during embryonic development.
Step-by-step explanation:
The structure that gives rise to the peripheral nervous system and certain facial structures is the neural crest cells. During embryonic development, as the neural folds come together, the underlying structure forms the neural tube, which gives rise to the central nervous system (CNS). Meanwhile, the cells detaching from the neural folds form the neural crest, which differentiates into many structures of the peripheral nervous system (PNS) and some non-nervous tissues such as craniofacial cartilage and bone. This includes a wide array of diverse cell types, including sensory and autonomic neurons, glial cells, melanocytes and more.