reel (n.1) Look up reel at Dictionary.com
"frame turning on an axis," especially one on which thread is wound, late Old English hreol "reel for winding thread," from Proto-Germanic *hrehulaz; probably related to hrægel "garment," and Old Norse hræll "spindle," from PIE *krek- "to weave, beat" (cognates: Greek krokus "nap of cloth").

Specifically of the fishing rod attachment from 1726; of a film projector apparatus from 1896. Reel-to-reel type of tape deck is attested from 1958.
reel (n.2) Look up reel at Dictionary.com
"lively Highland dance," 1580s, probably a special use of reel (n.1), which had a secondary sense of "a whirl, whirling movement" (1570s) or from reel (v.1). Applied to the music for such a dance from 1590s.
reel (v.1) Look up reel at Dictionary.com
"to whirl around," late 14c., also "sway, swing, rock, become unsteady" (late 14c.), "stagger as a result of a blow, etc." (c.1400), probably from reel (n.1), on notion of "spinning." Of the mind, from 1796. Related: Reeled; reeling.
reel (v.2) Look up reel at Dictionary.com
"to wind on a reel," late 14c., from reel (n.1). Verbal phrase reel off "recite without pause or effort" is from 1837. Fishing sense is from 1849. Related: Reeled; reeling.