[circular band] Old English hring "circlet of metal, especially one of a precious metal for wearing on the finger ornamentally, also a part of a mail coat; anything circular," from Proto-Germanic *hringaz "something curved, circle" (source also of Old Norse hringr, Old Frisian hring, Danish, Swedish, Dutch ring, Old High German hring, German Ring), from nasalized form of PIE root *sker- (2) "to turn, bend."
Other Old English senses were "circular group of persons" (birds, actually), also "horizon." In Old and Middle English also "a bracelet, armlet." As a token of marriage, betrothal, chastity, etc., by c. 1200. The sense of "a number of things arranged in a circle" is by 1580s.
The meaning "place for prize fight and wrestling bouts" (early 14c.) is from the space in a circle of bystanders in the midst of which such contests once were held, "... a circle formed for boxers, wrestlers, and cudgel players, by a man styled Vinegar; who, with his hat before his eyes, goes round the circle, striking at random with his whip to prevent the populace from crowding in" [Grose, 1785]. Hence the ring "boxing" (by 1770). The meaning "combination of persons interested in attaining some object" is from 1829, originally commercial or political, latter in reference to espionage or terrorism. Of the annual growth bands in trees, from 1670s.
Fairy ring is from 1620s. Ring finger, "third finger of the left hand" (in anatomy, of either hand) is Old English hringfingr, a compound also attested in other Germanic languages; it is also called ring-man (15c.). To run rings round (someone) "be superior to" is from 1891.
The nursery rhyme ring a ring a rosie is attested in an American form (with a different ending) from c. 1790. "The belief that the rhyme originated with the Great Plague is now almost universal, but has no evidence to support it and is almost certainly nonsense" ["Oxford Dictionary of English Folklore"]. This proposal of connection dates only to the late 1960s.