"To be or remain in that posture in which the weight of the body rests upon the posteriors" [OED], Middle English sitten, from Old English sittan "occupy a seat, be seated, sit down, seat oneself; remain, continue; settle, encamp; lie in wait; besiege" (class V strong verb; past tense sæt, past participle seten), from Proto-Germanic *setjan (source also of Old Saxon sittian, Old Norse sitja, Danish sidde, Old Frisian sitta, Middle Dutch sitten, Dutch zitten, Old High German sizzan, German sitzen, Gothic sitan), from PIE root *sed- (1) "to sit."
With past tense sat (formerly also set, which is now restricted to dialect, and sate, now archaic); and past participle sat, formerly sitten. The meaning "to have a seat in a legislative assembly" is from late 14c.; in reference to the assembly, "to hold a session," from 1510s. The sense of "pose" for a portrait, etc., is by 1530s.
As short for babysit (v.) by 1966. The specific sense of "occupy a judicial seat" (Old English) is the notion in sit in judgement. The meaning "have a certain position or direction" is from c. 1200; of winds, "to blow from" (a certain quarter), 1590s, from the notion of "to be in."
To sit back "be inactive" is from 1943. To sit on one's hands originally was "withhold applause" (1926), later generalized to "do nothing" (by 1959). To sit around "be idle, do nothing" is by 1915, American English. To sit out "not take part, make oneself an exception" is from 1650s.