Priapus was the Greek (and later, Roman) god of fertility. He was usually depicted as a satyr with a huge penis. Priapism is a medical condition characterized by an insistent erection (longer than four hours) in the absence of sexual stimulation. It is actually quite painful and not at all as enjoyable as the symptoms might suggest. If not treated promptly, the penis could be irreperably damaged.
Taqu coined the wonderful sniglet inappropriapism to describe those moments when you get an erection in circumstances that are really embarassing.