This is a novelty song based on a medley of traditional Mexican melodies. The Jarabe Tapatío was put together by Jesús González Rubio (c1804-74) who was Professor of Music at Guadalajara.
The parody was recorded by the American comedian and song parodist Allan Sherman (1924-73) who adapted it with his collaborator Lou Busch; a live recording was released in the UK on the Warner Bros label in November 1963 as the B Side of "Won't You Come Home, Disraeli?"
As the melody was already quite well known in the English speaking world, many people appear to have believed it to have been an authentic song rather than a parody.
Suggestion credit: Alexander Baron - London, England, for above 2