Middle English bough, from Old English bog "shoulder, arm," extended in Old English to "twig, branch of a tree" (compare limb (n.1)), from Proto-Germanic *bogaz (source also of Old Norse bogr "shoulder," Old High German buog "upper part of the arm or leg," German Bug "shoulder, hock, joint"), from PIE root *bhagu- "arm" (source also of Sanskrit bahus "arm," Armenian bazuk, Greek pakhys "forearm"). The "limb of a tree" sense is peculiar to English.
"ornamental ring or clasped chain for the wrist," mid-15c., from Old French bracelet (14c.), diminutive of bracel, from Latin bracchiale "armlet," from bracchium "an arm, a forearm," from Greek brakhion "an arm" (see brachio-).
"arm of the sea, estuary of a river," early 15c., Scottish, from Old Norse fjörðr (see fjord).
"ornamental ring worn upon the arm or ankle," 1787, from Hindi bangri "colored glass bracelet or anklet."
"pertaining to the armpit or shoulder," 1610s, from Latin *axillaris, from axilla "armpit, upper arm, underpart of an upper wing" (see axle).
"having wings, winged," 1660s, from Latin alatus, from ala "wing, armpit, wing of an army," from *axla, originally "joint of the wing or arm;" from PIE *aks- "axis" (see axis).
early 14c., "piece of armor for the arms," also "thong, strap for fastening," from Old French brace "arms," also "length measured by two arms" (12c., Modern French bras "arm, power;" brasse "fathom, armful, breaststroke"), from Latin bracchia, plural of bracchium "an arm, a forearm," from Greek brakhion "an arm" (see brachio-).
The meaning "that which holds two or more things firmly together" (on the notion of clasping arms) is from mid-15c. Hence the word is applied to various devices for fastening and tightening. The meaning "a prop, support," especially in architecture, is from 1520s. Of dogs, ducks, pistols, etc., "a couple, a pair" from c. 1400.
Braces is from 1798 as "straps passing over the shoulders to hold up the trousers;" from 1945 as "wires for straightening the teeth."