"an organized group," originally especially of armed men, late 15c., from 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. One-man band is by 1931 as "man who plays several musical instruments simultaneously;" figurative extension is by 1938.
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."