Old English hring "circular band," from Proto-Germanic *khrengaz (cf. Old Norse hringr, Old Frisian hring, German Ring), literally "something curved," from PIE *skrengh- nasalized form of (s)kregh-, from root *(s)ker- "to turn, bend," with wide-ranging derivative senses (cf. Latin curvus "bent, curved," crispus "curly;" Old Church Slavonic kragu "circle," and perhaps Greek kirkos "ring," koronos "curved").
Meaning "place for prize fight and wrestling bouts" (early 14c.) is from the space in a circle of bystanders in which such contests were once held (ringside is attested from 1866). Meaning "combination of interested persons" is from 1829. Tree ring is from 1670s; fairy ring is from 1620s. 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 connection only dates to the 1960s.
"sound a bell," Old English hringan, from Proto-Germanic *khrenganan (cf. Old Norse hringja, Swedish ringa, Middle Dutch ringen), probably of imitative origin. To give (someone) a ring "call on the telephone" was in use by 1910. To ring down a theatrical curtain is from 1772, from the custom of signaling for it by ringing a bell.