**Core Concept**
The nail bed of the index finger is supplied by a specific branch of the radial nerve, which is a branch of the brachial plexus. The brachial plexus is a complex network of nerves that originates from the spinal cord and supplies the upper limb.

**Why the Correct Answer is Right**
The correct answer is the lateral terminal branch of the radial nerve, which is responsible for supplying the skin of the back of the hand and the nail bed of the index and middle fingers. This branch arises from the radial nerve as it passes through the supinator muscle and continues distally to supply the skin and muscles of the forearm and hand.

**Why Each Wrong Option is Incorrect**
**Option A:** This option is incorrect because the median nerve primarily supplies the thumb, index, and middle fingers, but it does not supply the nail bed of the index finger.
**Option B:** This option is incorrect because the ulnar nerve supplies the little finger and ring finger, but not the index finger.
**Option C:** This option is incorrect because the musculocutaneous nerve primarily supplies the biceps brachii and brachialis muscles, but not the skin of the hand or fingers.

**Clinical Pearl / High-Yield Fact**
The brachial plexus is formed by the C5-C8 spinal nerves and the T1 spinal nerve, which merge to form the trunks and divisions of the brachial plexus. This complex network of nerves is responsible for supplying the muscles and skin of the upper limb.

**Correct Answer:** C. The lateral terminal branch of the radial nerve supplies the nail bed of the index finger.