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

The elven realm ruled by Galadriel and Celeborn is Lothlórien. Also called Lórien or the Golden Wood, it lies between the Misty Mountains and the River Anduin in eastern Middle-earth.

Lothlórien is home to the Galadhrim, Silvan Elves whose culture was shaped by Sindarin influence. Galadriel and Celeborn ruled from Caras Galadhon, a remarkable city built among the branches of tall mallorn trees. Galadriel helped bring the golden mallorn to the region, transforming its appearance and helping preserve its beauty.

The realm was protected by Galadriel's Ring of Power, Nenya. Its power resisted decay and helped shield Lothlórien from Sauron's influence and gaze. The Fellowship entered the forest after escaping Moria and stayed there for what felt like only a few days, although roughly a month passed.

Lothlórien is distinct from Rivendell, Elrond's refuge; Mirkwood, the woodland realm associated with Thranduil; and the Grey Havens, the Elven ports on Middle-earth's western coast.