The Wood of Lothlorien [also called Lorien] is a very tranquil and enchanting place. Its name means ‘Lorien of the blossom’. It was founded in the Third Age by Lady Galadriel and Lord Celeborn. Its location is the same as the ancient realm of Doriath, the domain of King Thingol and Melian in the First Age. Lothlorien was dedicated to assisting the forces that opposed Sauron in any way. In the Third Age, it was one of the very few elven realms remaining and it was the one that best preserved the beauty of Valinor, possibly the working of Galadriel who desired Valinor above all things. It’s chief city was Caras Galadhon, the heart of elvendom on earth.
One of the very important occurrences here was the second meeting of Arwen Undomiel and Aragorn Elessar upon the hill of Cerin Amroth. Arwen was staying with her grandmother, Galadriel for a short time and Aragorn wandered to Lothlorien by accident. Here, they pledged their love to each other and Arwen swore to bind herself to him. Aragorn soon left again for the wild, and several years later they met again in Rivendell for the Council of Elrond.
In The Fellowship of the Ring, the fellowship stays in Lothlorien for a short time. This realm was most likely abandoned when Lady Galadriel departed for Valinor but, because the fate of Celeborn is untold, he may have remained to govern any elves that decided against taking the ships to the Uttermost West.

++ References
The Silmarillion by J.R.R. Tolkien
The Lord of the Rings by J.R.R. Tolkien
The Unfinished Tales of Arwen and Aragorn by J.R.R. Tolkien