Anórien is a region and fiefdom of Gondor, called Sunlending by the Rohirrim. The name does not refer to the West but is 'heraldic', related to the name and the emblem of Anárion, son of Elendil: the land is immediately attached to the Minas Anor.
Anórien lay north of Minas Tirith and the line of the White Mountains, and was the only part of the northern half of the realm, Calenardhon, which was not given to the Éothéod to become the Kingdom of Rohan.
Anórien formed a narrow strip of land consisting of the valleys of the White Mountains, and its borders were the Mering Stream in the west, and the Mouths of the Entwash in the north. Its eastern border was the border of Gondor at the Anduin.
The name is purely heraldic and doesn't refer to its climate or sunshine.