c. 1600, "area in an ancient theater for the chorus," from Latin orchestra, from Greek orkhēstra, semicircular space where the chorus of dancers performed, with suffix -tra denoting place + orkheisthai "to dance," perhaps an intensive of erkhesthai "to go, come," but Beekes rejects the suggested derivations.
The PIE root is also the source of Sanskrit rghayati "trembles, rages, raves," rnoti "rises, moves," arnah "welling stream;" Old Persian rasatiy "he comes;" Hittite arai- "to rise, arise; lift, raise," hardu- "brood, descendance;" Armenian y-arnem "to rise;" Greek ornynai "to rouse, start;" Latin oriri "to rise," origo "a beginning;" Gothic rinnan, Old English irnan "to flow, run."
In ancient Rome, orchestra referred to the place in the theater reserved for senators and other dignitaries. Meaning "group of musicians performing at a concert, opera, etc." is recorded by 1720, so called because they occupy the position of the orchestra relative to the stage ; that of "part of theater in front of the stage" is from 1768 in English.
Some related words still retain the "dancing" sense: Orchestic "of or pertaining to dancing" (1712), also orchestric (1740).