band (v.) Look up band at Dictionary.com
1520s, "to bind or fasten;" also "to join in a company," from band (n.1) and (n.2) in various senses, and partly from French bander "to bind," from bande "a strip." The meaning "affix an ID band to (a wild animal, etc.)" is attested from 1914. Related: Banded; banding.
band (n.2) Look up band at Dictionary.com
"an organized group," originally especially of armed men, late 15c., from Middle French bande, which is traceable to the Proto-Germanic root of band (n.1), perhaps via a band of cloth worn as a mark of identification by a group of soldiers or others (compare Gothic bandwa "a sign"). But perhaps from Middle English band, bond in the sense "force that unites, bond, tie" (late 14c.). Also compare Old Norse band "cord that binds; act of binding," also "confederacy."

The extension to "group of musicians" is c. 1660, originally musicians attached to a regiment of the army and playing instruments which may be used while marching. To beat the band (1897) is to make enough noise to drown it out, hence to exceed everything.
band (n.1) Look up band at Dictionary.com
"a flat strip," also "something that binds," Middle English bende, from Old English bend "bond, fetter, shackle, chain, that by which someone or something is bound; ribbon, ornament, chaplet, crown," with later senses and spelling from cognate Old Norse band and technical senses from Old French bande "strip, edge, side" (12c., Old North French bende), all three ultimately from Proto-Germanic *bindan, from PIE root *bhendh- "to bind."

The meaning "a flat strip" (late 14c.) is from French. In Middle English, this was sometimes distinguished by the spelling bande, bonde, but with loss of terminal -e the words have fully merged via the notion of "flat strip of flexible material used to wind around something."

Meaning "broad stripe of color, ray of colored light" is from late 14c.; the electronics sense of "range of frequencies or wavelengths" is from 1922. Most of the figurative senses ("legal or moral commitment; captivity, imprisonment," etc.) have passed into bond (n.), which originally was a phonetic variant of this band. The Middle English form of the word is retained in heraldic bend (n.2) "broad diagonal stripe on a coat-of-arms."