It is not exactly made clear who Celeborn was in The Silmarillion, as Tolkien never truely answered the question during his life. His last writings indicate that Celeborn was not a Sindarin Elf, but rather a Teleri Elf from Valinor, then named Teleporno in Telerin, and travelled with Galadriel to Middle-earth, during the First Age. They left at the same time, but not as part of, the rebellion of the Noldor. Teleporno was "Sindarized" as Celeborn: Telerin t(y)eleb (silver) was translated to Sindarin keleb. If Celeborn's Aman origin is adopted, he must have been a part of the Telerin king Olwë's family, to be counted as a kinsman of the Sindarin king Thingol, Olwë's brother Elwë.
Celeborn remained behind after Galadriel left Middle-earth, but eventually, some time during the Fourth Age, left for Valinor. It is said with him passed the last memory of the Elder Days from Middle-earth.