The forest where the Elves live in The Hobbit is Mirkwood.
Mirkwood, originally called Greenwood the Great, lies east of the Misty Mountains. Its northern region is ruled by Thranduil, the Elvenking, whose people capture Thorin Oakenshield’s company after the travellers enter the forest. Bilbo avoids capture with the Ring and later helps the Dwarves escape in barrels down the Forest River.
The forest has been darkened by the influence of Dol Guldur and the Necromancer, who is later revealed to be Sauron. Tolkien’s earlier name, Greenwood the Great, remains important because it shows that the forest’s corruption is historical rather than inherent. Fangorn is a different forest inhabited by Ents, while Lothlórien is the woodland realm of Galadriel and Celeborn.