Priapus was a minor Greek fertility god best known for his large and permanently erect phallus. He was the son of Aphrodite, but there's some question as to whether his father was Pan, Zeus, Hermes, or one of Aphrodite's other numerous lovers. Priapus was a protector of gardens and orchards, and is typically portrayed as a homely old man with a raging erection.
According to legend, before his birth, Hera cursed Priapus with impotence as payback for Aphrodite's involvement in the whole Helen of Troy fiasco. Doomed to spend his life ugly and unloved, Priapus was tossed down to earth when the other gods refused to let him live on Mount Olympus.
He was raised by shepherds, and spent a lot of time hanging out with Pan and the satyrs. However, all this cavorting in the forest with the fertility spirits proved frustrating for Priapus, who remained impotent.
Eventually he tried to rape a nymph, but was thwarted when a braying donkey alerted her to his presence. He pursued the nymph, but the other gods helped her hide by turning her into a lotus plant.
In some stories, his lust left him with a permanent erection, and in others, he was punished by Zeus for the attempted rape by being bestowed a set of huge (but useless) wooden genitalia.
In the Greek countryside, Priapus was honored in homes and gardens, and doesn't appear to have had an organized cult following. He was seen as a protector deity in rural areas. In fact, statues of Priapus were often adorned with warnings, threatening trespassers, male and female alike, with acts of sexual violence as punishment.
His name gives us the medical term priapism, which is a condition in which a man can't get rid of his erection, despite a lack of stimulation, within four hours. It is actually considered a medical emergency.