Hyponychium
The hyponychium is the area of epithelium, particularly the thickened portion, underlying the free edge of the nail plate on the nail. Its proximal border is immediately distal to distal limit of nail bed—a.k.a. the onychodermal band. Its distal border the free edge of the nail plate—or where that edge would project downwards onto the epidermis.
It is sometimes called "the quick", as in the phrase cut to the quick.