After the War of Wrath, and at the beginning of the Second Age.
Many things were happening; Ships were being built, and the Grey Havens were established, and all the peoples of the west were on the move. Many decisions were being made.
At this time Celebrimbor Mastercrafter, Scion of the House of Fëanor, was kept very busy. It was he who built and designed the Havens and with the help of elves, men, and dwarves the work went quickly. Most helpful was the aid that came from the Noldo of the West and the were even Maia from Aulë who were there to clean up and to aid Eönwë as he was teaching the Edain, in preperation for their journey to Elanna, to establish the Kingdom of Númenor.
All who were leaving to go to Aman were busy and all who were staying were helping them get ready. Much was done and much was learned by all.
And it come to pass that the last ships were leaving for the west and Gil-galad came to Celebrimbor and asked him of his plans. Celebrimbor had not given this any thought because he had been so busy that he had had no time. Really he did not want it to end, but he had to make a choice. As he did not have to leave right away he choice to stay in Middle-earth. With the Havens complete, he traveled with the dwarves into the east to where they claimed a great city under the mountains awaited. This city was Hadhodrond, Khazad-dûm.
Khazad-dûm turned out to be a dream come true for Celebrimbor. The dwarves were masters of all forms of handicraft, and Celebrimbor was eager to learn. The dwarves also learned much from Celebrimbor. As he worked Celebrimbor gave thought to his one true dream; that was to surpass his grandfather Fëanor in all skills and greatness. As he did not have the inner fire to create as Fëanor his dream was doomed, but as long as he did not realize this all was well. Celebrimbor soon surpassed all his teachers and he started to teach all that he could.
From the depths of the mines of Khazad-dûm news started to come of greater and greater finds if mithil and gold in quanities that had never been heard of before. And the dwarves started to bring this mithril to Celebrimbor, for he was the mightiest in craft and mithril was the greatest of all metals light and strong.
Slowly over the years other elves heard of the works of Celebrimbor and they started to seek him out in Moria.