`Yet maybe this will lighten your heart,' said Galadriel; `for it was left in my care to be given to you, should you pass through this land.' Then she lifted from her lap a great stone of a clear green, set in a silver brooch that was wrought in the likeness of an eagle with outspread wings; and as she held it up the gem flashed like the sun shining through the leaves of spring. `This stone I gave to Celebrían my daughter, and she to hers; and now it comes to you as a token of hope. In this hour take the name that was foretold for you, Elessar, the Elfstone of the house of Elendil!'
Then Aragorn took the stone and pinned the brooch upon his breast, and those who saw him wondered; for they had not marked before how tall and kingly he stood, and it seemed to them that many years of toil had fallen from his shoulders.

(LoTR, II-8)

There was in Gondolin a jewel-smith named Enerdhil, the greatest of that craft among the Noldor after the death of Fëanor. Enerdhil loved all green things that grew, and his greatest joy was to see the sunlight through the leaves of trees. And it came into his heart to make a jewel within which the clear light of the sun should be imprisoned, but the jewel should be green as leaves. And he made this thing, and even the Noldor marvelled at it. For it is said that those who looked through this stone saw things that were withered or burned healed again or as they were in the grace of their youth, and that the hands of one who held it brought to all that they touched healing from hurt. This gem Enerdhil gave to Idril the King's daughter, and she wore it upon her breast; and so it was saved from the burning of Gondolin.
And before Idril set sail she said to Eärendil her son: "The Elessar I leave with thee, for there are grievous hurts to Middle-earth which thou maybe shalt heal. But to none other shalt thou deliver it."
And indeed at Sirion's Haven there were many hurts to heal both of Men and Elves, and of beasts that fled thither from the horror of the North; and while Eärendil dwelt there they were healed and prospered, and all things were for a while green and fair. But when Eärendil began his great voyages upon the Sea he wore the Elessar upon his breast, for amongst all his searchings the thought was always before him: that he might perhaps find Idril again; and his first memory of Middle-earth was the green stone above her breast, as she sang above his cradle while Gondolin was still in flower. So it was that the Elessar passed away, when Eärendil returned no more to Middle-earth.

In ages after there was again an Elessar, and of this two things are said, though which is true only those Wise could say who now are gone. For some say that the second was indeed only the first returned, by the grace of the Valar; and that Olórin (who was known in Middle-earth as Mithrandir) brought it with him out of the West. And on a time Olórin came to Galadriel, who dwelt now under the trees of Greenwood the Great; and they had long speech together. For the years of her exile began to lie heavy on the Lady of the Noldor, and she longed for news of her kin and for the blessed land of her birth, and yet was unwilling to forsake Middle-earth [this sentence was changed to read: but was not permitted yet to forsake Middle-earth]. And when Olórin had told her many tidings she sighed, and said:
"I grieve in Middle-earth, for leaves fall and flowers fade; and my heart yearns, remembering trees and grass that do not die. I would have these in my home." Then Olórin said: "Would you then have the Elessar?"
And Galadriel said: "Where now is the Stone of Eärendil? And Enerdhil is gone who made it."
"Who knows?" said Olórin. "Surely," said Galadriel, "they have passed over Sea as almost all fair things beside. And must Middle-earth then fade and perish for ever?" "That is its fate," said Olórin. "Yet for a little while that might be amended, if the Elessar should return. For a little until the Days of Men are come." "If and yet how could that be," said Galadriel. "For surely the Valar are now removed and Middle-earth is far from their thought, and all who cling to it are under a shadow."
"It is not so," said Olórin. "Their eyes are not dimmed nor their hearts hardened. In token of which look upon this!" And he held before her the Elessar, and she looked on it and wondered. And Olórin said: "This I bring to you from Yavanna. Use it as you may, and for a while you shall make the land of your dwelling the fairest place in Middle-earth. But it is not for you to possess. You shall hand it on when the time comes. For before you grow weary, and at last forsake Middle-earth one shall come who is to receive it, and his name shall be that of the stone: Elessar he shall be called."

The other tale runs so:
that long ago, ere Sauron deluded the smiths of Eregion, Galadriel came there, and she said to Celebrimbor, the chief of the Elven-smiths: "I am grieved in Middle-earth, for leaves fall and flowers fade that I have loved, so that the land of my dwelling is filled with regret that no Spring can redress."
"How otherwise can it be for the Eldar, if they cling to Middle-earth?" said Celebrimbor. "Will you then pass over Sea?"
"Nay," she said. "Angrod is gone, and Aegnor is gone, and Felagund is no more. Of Finarfin's children I am the last. But my heart is still proud. What wrong did the golden house of Finarfin do that I should ask the pardon of the Valar, or be content with an isle in the sea whose native land was Aman the Blessed? Here I am mightier."
"What would you then?" said Celebrimbor.
"I would have trees and grass about me that do not die here in the land that is mine," she answered. "What has become of the skill of the Eldar?" And Celebrimbor said: "Where now is the Stone of Eärendil? And Enerdhil who made it is gone." "They have passed over Sea," said Galadriel, "with almost all fair things else. But must then Middle-earth fade and perish for ever?"
"That is its fate, I deem," said Celebrimbor. "But you know that I love you (though you turned to Celeborn of the Trees), and for that love I will do what I can, if haply by my art your grief can be lessened." But he did not say to Galadriel that be himself was of Gondolin long ago, and a friend of Enerdhil, though his friend in most things outrivalled him. Yet if Enerdhil had not been then Celebrimbor would have been renowned. Therefore he took thought, and began a long delicate labour, and so for Galadriel he made the greatest of his works (save the Three Rings only). And it is said that more subtle and clear was the green gem that he made than that of Enerdhil, but yet its light had less power. For whereas that of Enerdhil was lit by the Sun in its youth, already many years had passed ere Celebrimbor began his work, and nowhere in Middle-earth was the light as clear as it had been, for though Morgoth had been thrust out into the Void and could not enter again, his far shadow lay upon it. Radiant nonetheless was the Elessar of Celebrimbor; and he set it within a great brooch of silver in the likeness of an eagle rising upon outspread wings." Wielding the Elessar all things grew fair about Galadriel, until the coming of the Shadow to the Forest. But afterwards when Nenya, chief of the Three," was sent to her by Celebrimbor, she needed it (as she thought) no more, and she gave it to Celebrian her daughter, and so it came to Arwen and to Aragorn who was called Elessar.

At the end is written:

The Elessar was made in Gondolin by Celebrimbor, and so came to Idril and so to Eärendil. But that passed away. But the second Elessar was made also by Celebrimbor in Eregion at the request of the Lady Galadriel (whom he loved), and it was not under the One, being made before Sauron rose again.

This narrative goes with "Concerning Galadriel and Celeborn" in certain features, and was probably written at about the same time, or a little earlier. Celebrimbor is here again a jewel-smith of Gondolin, rather than one of the Fëanorians (cf. p.247); and Galadriel is spoken of as being unwilling to forsake Middle-earth (cf. p.246) though the text was later emended and the conception of the ban introduced, and at a later point in the narrative she speaks of the pardon of the Valar.
Enerdhil appears in no other writing; and the concluding words of the text show that Celebrimbor was to displace him as the maker of the Elessar in Gondolin. Of Celebrimbor's love for Galadriel there is no trace elsewhere.
...
Lastly, it may be remarked that the healing power here ascribed to the Elessar at the Havens of Sirion is in The Silmarillion (p. 247) attributed to the Silmaril.

(UT, THE HISTORY OF GALADRIEL AND CELEBORN and of Amroth King of Lórien; Elessar)