Anarríma (anar = sun) was one of the constellations set in the heavens by Varda to enlighten the awakening of the elves and gathered by Varda from among the ancient stars (The Silmarillion, Chap. 3; cf. MR 71, 160). Other constellations were Wilwarin, Telumendil, Soronúmë, and as the most important ones, Menelmacar (or Telumehtar) and Valacirca.
The elements of the name seem to be anar 'sun', and ríma 'edge, hem, border' (cf. e.g. Etymologies s.v. ANÁR, RÎ; LR 348, 383), thus: 'Sun-border'?
It is not clear (to me, at least) what is the name of the constellation in "modern" terms, but it should be one observable in the northern hemisphere. The word ríma might refer to the Corona Borealis (the Northern Crown) but one might also consider the Great Square of Pegasus, easily recognized from its four bright stars.
Anarríma entered the mythology only in The Later Quenta Silmarillion, composed after the completion of The Lord of the Rings, in early 1950's (cf. MR 141). In the previous versions of the story, only the Great Bear (or Valacirca) and Orion (or Telumehtar / Menelmacar with a curious history of its own) are mentioned.
The printed text in The Silmarillion follows quite closely this late version (MR 160). The only notable difference concerning the constellations (and among others, Anarríma) is that in The Silmarillion it is only told that Varda set the constellations "as signs in the heavens of Arda" (Silmarillion, Chap. 3), but in The Later Quenta Silmarillion text J.R.R. Tolkien writes: 'as signs in Heaven that the gods may read' (MR 160).
LR = The Lost Road and other Writings (The History of Middle-earth, vol. 5)
MR = Morgoth's Ring (The History of Middle-earth, vol. 10)