1640s, "course, direction," from Latin ductus "a leading, a conduit pipe," noun use of past participle of ducere "to lead," from PIE root *deuk- "to lead." Anatomical sense "vessel of an animal body by which blood, lymph, etc., are conveyed" is from 1660s. Meaning "conduit, channel" is 1713; that of "air tube in a structure" is from 1884.
Duct tape originally was duck tape (1894), long, non-adhesive strips of plain cotton duck cloth used in various mechanical processes; from duck (n.2). The name was transferred to a plastic-coated adhesive tape used by U.S. soldiers in World War II (perhaps in part because it was waterproof). It continued in civilian use after the war, and the name shifted to duct tape by 1958, perhaps because it was frequently used on air ducts, which also accounts for its standard silver-gray color.