Wordorigins.org

A gargoyle is a stone figure that forms part of the gutter system of medieval cathedrals. It is a spout, in the shape of some grotesque creature, that carries rainwater away from the walls of the building. The word is from the Old French gargouille, or throat, and is a reference to the water passing through the throat of the stone figure. The same root gives us the modern noun and verb gargle.

From the 14th century poem St. Erkenwald:

Hit was a throghe of thykke stone...With gargeles garnysht aboute, all of gray marbre.
(it was a tomb of thick stone…with gargoyles garnished about, all of gray marble.)

According to myth, in the 7th century a dragon, named Gargouille rose from the waters of the Seine River in France. Unlike the typical dragons of mythology, this one did not breathe fire, but rather was a water dragon. The monster proceeded to lay waste to the countryside around Paris by drowning it. St. Romain, the Archbishop of Rouen, accompanied only by a condemned prisoner, set out to stop the beast. Upon confronting the monster, the saint formed a cross with his two index fingers, taming Gargouille. The dragon was led back to Paris, where it was slain and burned. The head, however, was saved and mounted on a building. This legend supposedly gave rise to the architectural practice of designing waterspouts to look like monsters.

(Source: Oxford English Dictionary, 2nd Edition)