Hunc lucum tibi dedico, consecroque, Priape,
Qua domus tua Lampsaci est, quaque silva, Priape,
Nam te praecipue in suis urbibus colit ora
Hellespontia, caeteris ostreosior oris.

This grove I dedicate and consecrate to thee, Priapus, who hast thy home at Lampsacus, and eke thy woodlands, Priapus; for thee especially in its cities worships the coast of the Hellespont, richer in oysters than all other shores.
Caius Valerius Catullus, XVIII.

Fresco, 1st century AD, at the Villa dei Vetii, Pompeii

Priapos isn’t mentioned in the works of the earliest Greek poets, such as Homer and Hesiod ^{[ 1 ]}; like many of the Greek and Roman pantheon, he was an import from Asia Minor. His cult wasn’t introduced in Greece until around 400 BCE and he was never as popular with the Greeks as he became with the Romans. His original cult centre was around the Hellespont, especially in Lampsakos in Mysia ^{[ 2 ]} where he was said to have been born, and the city of Priapus on the south coast of the Propontis which derived its name from the most important god of the local pantheon. Roman depictions of Priapos wearing a peaked Phrygian cap – for example the famous fresco of the 1st century AD from the Villa dei Vetii, Pompeii, in which he is seeing favourably weighing his phallus against a bag full of money – reflect the god’s Anatolian origins. 

In Asia Minor, the identity of Priapos seems to have been combined with that of Pan, the Greek god of woods and fields, with whom he is depicted on coins from Nicaea. But he was more than simply a rustic fertility god with a prodigiously large phallus: Lucian relates that in Bithynia he was accounted a warlike god, the tutor of Ares, teaching the young battle god to dance before teaching him to be a warrior:

"Attention is next claimed by the Roman dance of the Salii, a priesthood drawn from the noblest families; the dance is performed in honour of Mars, the most warlike of the Gods, and is of a particularly solemn and sacred character. According to a Bithynian legend, which agrees well with this Italian institution, Priapus, a war-like divinity (probably one of the Titans, or of the Idaean Dactyls, whose profession it was to teach the use of arms), was entrusted by Hera with the care of her son Ares, who even in childhood was remarkable for his courage and ferocity. Priapus would not put weapons into his hands till he had turned him out a perfect dancer; and he was rewarded by Hera with a tenth part of all Ares's spoils." ^{[ 3 ]}

References:

[1] "Priapos [Lampsakos] is a city on the sea, and also a harbor. Some say that it was founded by Milesians . . . It was named after Priapos, who was worshipped there; then his worship was transferred thither from Orneai near Korinthos, or else the inhabitants felt an impulse to worship the god because he was called the son of Dionysos and a Nymphe; for their country is abundantly supplied with the vine, both theirs and the countries which border next upon it, I mean those of the Parianoi and the Lampsakenoi. At any rate, Xerxes gave Lampsakos to Themistokles to supply him with wine. But it was by people of later times that Priapos was declared a god, for even Hesiod does not know of him; and he resembles the Attic deities Orthane, Konisalos, Tykhon, and others like them." Strabo, Geography 13. 1. 12

[2] “Among the people of Lampsacus, Priapus, who is the same as Dionysus, is held in honour and has the by-name Dionysus as well as Thriambus and Dithyrambus.”  Athenaeus, Deipnosophistae 1. 30b [source]
"This god [Priapos] is worshipped where goats and sheep pasture or there are swarms of bees; but by the people of Lampsakos he is more revered than any other god, being called by them a son of Dionysos and Aphrodite." Pausanias, Description of Greece 9. 31. 2 

[3] Lucan, De Saltatione, 20-22 [source]