Short answers, pulled from the story.
Galadriel and Celeborn rule Lothlórien during the Third Age, from their city of Caras Galadhon. The wood-elves of the realm are called the Galadhrim. Galadriel carried one of the Three Elf-Rings and used it to prevent Sauron from perceiving what lay within the realm.
Lothlórien is a Sindarin name meaning Dreamflower, used during the Third Age. Earlier names for the same forest include Lórinand (Valley of Gold) and Lindórinand (Valley of the Land of the Singers), the oldest Nandorin name. The people of Rohan called it Dwimordene, from Old English words for illusion and valley.
The Fellowship of the Ring spent roughly a month in Lothlórien but felt only days had passed. Sam Gamgee noticed the moon had gone from waning to new during their stay. The characters offer competing explanations: Frodo believed the realm existed in a different epoch entirely, Legolas attributed it to the Elves' different perception of passing time, and Aragorn suggested the Fellowship felt time as Elves do while inside the realm.
Caras Galadhon is the city of Lothlórien and the main settlement of the Galadhrim in Middle-earth. Founded by Amroth in the Third Age, it stands roughly ten miles from where the rivers Silverlode and Anduin converge. Its dwellings and guard-posts are built on platforms atop tall mallorn trees, lit at night by lamps burning green, gold, and silver.
Tom Shippey argued that the word Angle, used by the elf Haldir for the triangular heartland between the rivers, connects Lothlórien to old England by way of the Anglo-Saxon homeland near the Flensburg Fjord and the River Schlei. Verlyn Flieger analyzed the time paradoxes in detail, comparing them to the medieval tale Thomas the Rhymer and the Danish ballad Elvehøj. Paul H. Kocher, Elizabeth Danna, and Susan Robbins each explored the realm as a figure of biblical light striving against the darkness of Sauron.
Peter Jackson based the visual design of Lothlórien on the artwork of the conceptual designer Alan Lee. Some Lothlórien scenes were filmed on location in Paradise Valley near Glenorchy, New Zealand.