The medical term for an inflammation of the nail bed is onychia.

Onychia

An inflammation of the nail bed resulting in pain, swelling, and sometimes loss of the nail.

Nail Bed

The skin beneath the nail plate that provides nourishment and support for nail growth.

Inflammation

A biological response of body tissues to harmful stimuli, such as pathogens, damaged cells, or irritants, characterized by redness, warmth, swelling, and pain.