For about three hundred years, a vaudeville was a song. Only in the 1870s did American theater managers turn it into a kind of show.
The traditional origin lies in Normandy. Vire is a town in Calvados, on a river of the same name. In the 15th century a poet of Vire named Olivier Basselin, described in the OED as a fuller, was credited with satirical popular songs. They became known as songs of the Vau de Vire, the Vire valley. Early in the 17th century a Norman lawyer, Jean Le Houx, published a collection of such songs. Wikipedia notes he very likely wrote the ones credited to Basselin himself.
According to Etymonline, French vaudevire was altered to vaudeville under the influence of ville, "town," and the French word dates from the 16th century. The OED repeats the Vire story, but hedges: the name "is said to have been" first given to Basselin's songs.
Not everyone accepts the valley. A popular alternative derives the word from voix de ville, "voice of the town," for the topical songs of city streets. Wikipedia notes the origin "has yet to be determined." The French etymologist P. Guiraud, author of a 1982 dictionary of obscure etymologies, called place-name stories of this kind mostly apocryphal. He proposed a compound of the Norman verbs vauder and virer, "turn," describing a song's refrain.
What is clear is the path of the meaning. English recorded vaudeville in 1735 as a popular song, especially one sung on stage. In France the songs fed a theatrical form, the comédie en vaudevilles, which from the late 17th century mixed acting with songs.
The American sense came out of a marketing problem. Variety shows grew out of saloons and were seen as rough. From the 1860s the New York manager Tony Pastor courted middle-class audiences with cleaner programs. He banned liquor and bawdy material, and in early 1876 he began calling his shows vaudeville instead of variety. The French word sounded respectable, and other managers followed.
Etymonline dates the American variety-show sense to 1881. Cinema then eroded it from the 1910s. When New York's Palace Theatre switched to films only in November 1932, many treated it as the end of vaudeville.