The Galadhrim were the Silvan Elves that inhabited the woods of Lothlórien. Some of the Nandor, the Telerin Elves who abandoned the march of the Eldar on the eastern side of the Misty Mountains settled down in the Vales of Anduin and later on established the realms of Mirkwood in the north and Lórien in the south of Greenwood the Great.
In the beginning the Elves of Lórien were a small and scattered people, but after the War of Wrath at the beginning of the Second Age many of the Sindar of Beleriand migrated eastward. The Silvan Elves of Lórien welcomed them. The coming of the Sindar was a great boost to the power and culture of the Silvan Elves. The Silvan language was replaced by Sindarin.
The arrival of one in particular affected the Nandor of Laurelindórenan: the coming of a Sinda named Amdír. He took over Laurelindórenan. Under his leadership the people of Lórien prospered and their numbers grew. But Amdir was slain in the Battle of Dagorlad along with most of the army of Lórien. And Amroth his son grew weary of Middle-earth and wished to go to Valinor. But he drowned in the Bay of Belfalas.
In her wisdom Galadriel saw that Lórien would be a stronghold and a point of power to prevent the shadow from crossing the Anduin, for the Galadhrim were hardy and valiant. But they needed a ruler of greater strength and wisdom than they possessed. That is why Galadriel and Celeborn took up their permanent abode in Lórien and its government. Under the leadership of Galadriel and Celeborn the Galadhrim prospered and their skill grew. They became proficient in making Lembas, Elven rope, Elven-cloaks and many other crafts.
The Galadhrim defended their home against orcs, who attacked around the end of the Third Age, just before Sauron was destroyed. The Galadhrim weathered all of the attacks. For a more complete history of these battles, see Lothlórien.