A 'glee' is defined in Webster's Collegiate Dictionary as a part song for usually male voices. They are generally high-spirited, short, joyful songs. The posts by Highlandman and others who have consulted their reference works are correct on the form.
I have several 19th c. songbooks including Carmina Princetonia and the Academy songbook, which contain glees; songs which were the meat for Glee clubs. Abby is essentially correct on the modern usage.

The term is not exclusively 'English,' but was used wherever English is spoken.