1540s, "the common daw," a type of small European crow (Corvus monedula), "which frequents church towers, old buildings, etc.; noted for its loquacity and thievish propensities" [OED]. See jack (n.) + daw.
In modern times, parrots are almost the only birds that have the gift of speech, though connoisseurs are not ignorant that starlings and jackdaws have good abilities in that way, when properly educated. ["Chambers' Home Book and Pocket Miscellany," 1853]
In U.S. sometimes applied to a species of grackle.