When you let a fingernail grow (usually the thumb or index finger) so it's just long enough to perform handy tasks.
This can be useful for opening packages, scraping stuff, picking your nose (if you're so inclined), self defense, plucking a stringed instrument, itching scratches... etc.
Guy 1: Hey Bill, do you need some scissors to open that box?
Bill: That's ok, I have my utility nail.
Bill proceeds to puncture the plastic with his utility nail then rip open the rest of the box.
Sue noticed a spot of dried crud on her project just before turning it in. Luckily, she had her utility nail and was able to scrape it off.