mid-15c., from Old Norse draga, or a dialectal variant of Old English dragan "to draw," both ultimately from Proto-Germanic *dragan "to draw, pull," from PIE root *dhragh- "to draw, drag on the ground" (cf. Sanskrit dhrajati "pulls, slides in," Russian drogi "wagon;" but not considered to be directly the source of Latin trahere).
Meaning "to take a puff" (of a cigarette, etc.) is from 1914. Related: Dragged; dragging. Drag-out "violent fight" is from c.1859. To drag (one's) feet (1946, in figurative sense) supposedly is from logging, from a lazy way to use a two-man saw.