hangnail

Short definition from Wiktionary

Noun

1. 1
   A loose, narrow strip of nail tissue protruding from the side edge and anchored near the base of a fingernail or toenail.
2. 2
   A pointed upper corner of the toenail (often created by improperly trimming by rounding the corner) that, as the nail grows, presses into the flesh or protrudes so that it may catch (“hang”) on stockings or shoes.