I have a throbbing pain under my fingernail that came all of a sudden. It hurts when I press on it, and also hurts just sitting there. What is this?

Unknown. If there is pain it usually indicates either an injury or possibly a disease process as an infection or growth. Best to have it checked out.
Probable glomus tumo. This sounds as if it were a glomus rumor of the nailbed. Look it up here: http://emedicine.Medscape.Com/article/1255586-overview.