**Core Concept**
The 2nd branchial arch, also known as the hyoid arch, plays a crucial role in the development of the face and neck. It is associated with the formation of several nerves that innervate the muscles and provide sensory innervation to the skin of the neck.

**Why the Correct Answer is Right**
The nerve of the 2nd branchial arch is the facial nerve (cranial nerve VII). This nerve arises from the brainstem and passes through the stylomastoid foramen, providing motor innervation to the muscles of facial expression, as well as sensory innervation to the skin of the face and anterior two-thirds of the tongue. The facial nerve also has a branch called the chorda tympani, which carries taste fibers from the anterior two-thirds of the tongue.

**Why Each Wrong Option is Incorrect**
**Option A:** The glossopharyngeal nerve (cranial nerve IX) is associated with the 3rd branchial arch and provides sensory innervation to the posterior one-third of the tongue, as well as motor innervation to the stylopharyngeus muscle.
**Option B:** The vagus nerve (cranial nerve X) is a mixed nerve that arises from the brainstem and provides motor, sensory, and parasympathetic innervation to various organs in the neck and thorax.
**Option C:** The spinal accessory nerve (cranial nerve XI) is a motor nerve that arises from the brainstem and provides innervation to the sternocleidomastoid and trapezius muscles.

**Clinical Pearl / High-Yield Fact**
The facial nerve is often tested in clinical exams by assessing the patient's ability to identify the taste of sweet, sour, salty, and bitter substances on the anterior two-thirds of the tongue.

**Correct Answer: C. The facial nerve.