"circular band," Old English hring "small circlet, especially one of metal for wearing on the finger or as 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," also "horizon." 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]. Meaning "combination of interested persons" is from 1829. Of trees, from 1670s; fairy ring is from 1620s. Ring finger is Old English hringfingr, a compound found in other Germanic languages. To run rings round (someone) "be superior to" is from 1891.
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.