"The Lady of Lórien! Galadriel!" cried Sam. "You should see her, indeed you should, sir. I am only a hobbit, and gardening's my job at home, sir, if you understand me, and I'm not much good at poetry - not at making it: a bit of comic rhyme, perhaps, now and again, you know, but not real poetry - so I can't tell you what I mean. It ought to be sung. You'd have to get Strider, Aragorn that is, or old Mr Bilbo, for that. But I wish I could make a song about her. Beautiful she is, sir! Lovely! Sometimes like a great tree in flower, sometimes like a white daffadowndilly, small and slender like. Hard as di'monds, soft as moonlight. Warm as sunlight, cold as frost in the stars. Proud and far-off as a snow-mountain, and as merry as any lass I ever saw with daisies in her hair in springtime. But that's a lot o' nonsense, and all wide of my mark."
LotR, Book IV, Chapter V - The Window on the West
Gallery
Galadriel, daughter of Finarfin of the Noldor and of Eärwen of the Teleri, sister of Finrod Felagund, was born in Valinor.
Following the customs of the Eldar in Valinor, she received two names: her father named her Artanis and her mother Nerwen. Later, Celeborn gave her an epessë, an "after-name": Altáriel (in Telerin: Alatáriel), of which she used the Sindarin translation, Galadriel.
Fëanor loved her, fascinated by her golden hair in which the light of the Two Trees, Laurelin and Telperion, was said to have been snared. Thrice he begged her a tress. Thrice she refused it, foreseeing in him a darkness that she hated and feared.
Nevertheless she followed the Noldor in exile, dreaming of founding her own kingdom in Middle-Earth. (Some texts mention her leaving Valinor separately, with Celeborn)
She wedded Celeborn during First Age. The Silmarillion discribes him as a Sinda, Elu Thingol's kinsfolk (in the Unfinished Tales, Celeborn is a Teleri from Alqualondë).
They lived a while in Doriath and, in the beginning of Second Age, they crossed Ered Lindon and dwelled in Eriador, on the banks of the lake Nenuial where they became known as the Lord and the Lady of Eriador. There, between 350 and 400 SA was born their son Amroth whom other tales make a Sinda without any family tie with them (See Nimrodel). Later she had a daughter, Celebrían, who married Elrond.
In 700 SA, Galadriel perceived the presence of an evil spirit East of Hithaeglin. Therefore they found the Noldorin Kingdom of Eregion and in 750, the City of Ost-in-Edhil, near Khazad-dûm.
Galadriel began to establish contacts with the Sylvan Elves of Lórinand and c. 1350-1400 she left Eregion with her children after Annatar (Sauron) had incited the Gwaith-i-Mírdain (the People of the Jewel-smiths) to revolt against them, and she went to Lórinand.
After the fall of Eregion, Celebrimbor gave her Nenya, the White Ring of Water.
Later she left Lórinand, leaving its lead to Amroth and, having found Celeborn in Imladris, she dwelled between the Gwathló and Ethir Anduin, in a place which would be called later Dol Amroth. She only came back in Lórinand, called in that time Lothlórien or Lórien, after the tragic death of Amroth and Nimrodel in 1981 Third Age.
Galadriel was member of the White Council - including, among others, the Istari(Gandalf, Saruman and Radagast being the most famous of them), Elrond, Círdan,... - which met for the first time in 2463 Third Age. Not trusting Saruman, she would have preferred Gandalf to lead the Council.
She welcomed the Fellowship of the Ring when they went through Lórien, testing each of them, promising them what they most desire if they forsook their quest.
About one month after their arrival, she showed Frodo and Sam the Mirror of Galadriel wherein it was possible to see events occuring elsewhere in space as well as in time. There she revealed Nenya to Frodo and refused the proposal Frodo made her to give her the One Ring she had desired for long years, hoping to use it to restore order and justice.
When the Fellowship left Lórien, she joined them on a swan-shaped boat and after a parting feast in Egladil, she offered gifts to each member of the Fellowship:
To Aragorn, a sheath overlaid with a tracery of flowers and leaves wrought of silver and gold, and on it were set in elven runes formed of many gems the name Anduril and the lineage of the sword. She also gave him the Elessar, a great stone of a clear green that was wrought in the likeness of an eagle with outspread wings. It had been created by Enerdhil of Gondolin and worn by Idril Celebrindal and Eärendil.
To Boromir, a belt of gold.
To Merry and Pippin, small silver belts with a clasp wrought as a golden flower.
To Legolas, a bow longer and stouter than the bows of Mirkwood.
To Sam, a box with a G rune upon the lid, wherein was earth from Galadriel's orchard and a Mallorn nut.
To Frodo, a small crystal phial in which was caught the light of Eärendil's star.
Gimli at first refused any gift, but pressed to make a wish, he answered:
There is nothing, Lady Galdriel, said Gimli, bowing low and stammering. Nothing, unless it might be - unless it is permitted to ask, nay, to name a single strand of your hair, which surpasses the gold of the earth as the stars surpass the gems of the mine. I do not ask for such a gift. But you commanded me to name my desire.
To the surprise of all the Elves, Galadriel acceded to his wish - she had refused Fëanor such a gift - and gave him three golden hair which later would be set in crystal as a pledge of good will between the Mountain and the Wood.
Galadriel attended Arwen and Aragorn's wedding.
In the beginning of Fourth Age, she went to the Grey Havens without Celeborn and, accompanied by Elrond, Glorfindel, Frodo and Gandalf, she crossed the sea and passed in the West from which she never came back (sob).
However that may be, we'll leave the final word to Gimli:
Gimli wept openly.
- I have looked the last upon that which was fairest, he said to Legolas his companion. Henceforward I will call nothing fair unless it be her gift.
He put his hand to his breast.
- Tell me, Legolas, why did I come on this Quest? Little did I know where the chief peril lay! Truly Elrond spoke, saying that we could not foresee what we might meet upon our road. Torment in the dark was the danger that I feared, and it did not hold me back. But I would not have come had I known the danger of light and joy. Now I have taken my worst wound in this parting, even if I were to go this night straight to the Dark Lord. Alas for Gimli son of Glóin!
You can listen to Galadriel's Lament here