The forest later called Mirkwood was originally known as Greenwood the Great.
In Tolkien’s legendarium, Greenwood the Great was a vast forest east of the Anduin. It was an important realm of the Silvan Elves, and Thranduil established his woodland kingdom there. After the rise of the Necromancer, Sauron, at Dol Guldur in the southern part of the forest, shadow and danger spread through the region.
The forest consequently became known as Mirkwood, a name that reflects its corruption and peril. The change is part of the history behind The Hobbit, where Bilbo and the Dwarves cross the dangerous woodland on their journey to Erebor.
Fangorn Forest is a different forest, associated with the Ent Treebeard, while Lothlórien is the Elven realm ruled by Galadriel and Celeborn. The Old Forest lies near the Shire. These settings are all wooded, but none is the earlier name of Mirkwood.