Meaning "that which holds two or more things firmly together" (on notion of clasping arms) is from mid-15c. Hence applied to various devices for fastening and tightening. 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."
It forms all or part of: abbreviate; abbreviation; abridge; amphibrach; brace; bracelet; brachio-; brachiopod; brachiosaurus; brachy-; brassiere; breviary; brevity; brief; brumal; brume; embrace; merry; mirth; pretzel; vambrace.
It is the hypothetical source of/evidence for its existence is provided by: Greek brakhys "short;" Latin brevis "short, low, little, shallow;" Old Church Slavonic bruzeja "shallow places, shoals;" Gothic gamaurgjan "to shorten."
"beer saloon in which food is served," 1864, from French brasserie "beer-garden attached to a brewery," from brasser "to brew," from Latin brace "grain used to prepare malt," said by Pliny to be a Celtic word (compare Welsh brag "malt").
early 14c., "post, pillar, or beam used for support," from Old French estanchon "prop, brace, support" (13c., Modern French étançon), probably from estant "upright," from present participle of ester "be upright, stand," from Latin stare "to stand" (from PIE root *sta- "to stand, make or be firm").