What is the name of the Elvish realm ruled by Galadriel and Celeborn?

The Elvish realm ruled by Galadriel and Celeborn is Lothlórien. It lies east of the Misty Mountains between the rivers Anduin and Silverlode and is known for its mallorn trees.

The Fellowship enters Lothlórien after escaping Moria. Galadriel and Celeborn receive the travelers, provide counsel, and give them supplies and boats for the journey down the Anduin. Galadriel also tests the visitors by showing them visions in her Mirror.

Lothlórien has several names in Tolkien’s writings, including Lórien and Laurelindórenan. These names reflect its long history and the changing languages of its inhabitants. The forest is portrayed as a place where the memory of the Elder Days remains unusually strong.

Rivendell is a separate Elvish refuge ruled by Elrond, while Mirkwood is the woodland realm associated with Thranduil. Gondolin was an ancient hidden city of the First Age, not the woodland realm visited by the Fellowship.