late 15c., from French béguine (13c.), Medieval Latin beguina, a member of a women's spiritual order said to have been founded c.1180 in Liege in the Low Countries. They are said to take their name from the surname of Lambert le Bègue "Lambert the Stammerer," a Liege priest who was instrumental in their founding, and it's likely the word was pejorative at first.
The order generally preserved its reputation, though it quickly drew imposters who did not; nonetheless it eventually was condemned as heretical. A male order, called Beghards founded communities by the 1220s in imitation of them, but they soon degenerated (cf. Old French beguin "(male) Beguin," also "hypocrite") and wandered begging in the guise of religion; they likely were the source of the words beg and beggar, though there is disagreement over whether Beghard produced Middle Dutch beggaert "mendicant" or was produced by it.
Cole Porter's "Begin the Beguine" (1935) refers to a kind of popular dance of West Indian origin, from French colloquial béguin "an infatuation, boyfriend, girlfriend," earlier "child's bonnet," and before that "nun's headdress" (14c.), from Middle Dutch beggaert, ultimately the same word.