Final answer:
The oculomotor nerve, also known as cranial nerve III, is responsible for controlling the movement of the eyeballs and eyelids, as well as adjusting the pupils and lens of the eye.
Step-by-step explanation:
The cranial nerve that moves the eyeballs/eyelids, and adjusts the pupils and lens of the eye is the oculomotor nerve, which is cranial nerve III. The oculomotor nerve is responsible for the innervation of most of the extraocular muscles, including the levator palpebrae superioris muscle that raises the upper eyelid. It plays a key role in coordinating eye movements, along with the trochlear nerve (IV) and the abducens nerve (VI), and is part of the system responsible for the vestibulo-ocular reflex, which stabilizes gaze during head movements.