Celeborn remained in Middle-earth when Galadriel sailed into the West at the end of the Third Age. He remained for some time in Lórien, alone, and then retired to Rivendell, where Elrond's sons still dwelt. This is the last we hear of him in detail, but there is good reason to think that he eventually followed Galadriel into the West.
When first devised, Celeborn's name was intended to mean 'Silver Tree' (and in fact he shares this name with an actual tree, Celeborn of Tol Eressëa). There are indications that Tolkien intended to change this interpretation, and it seems that his final preference was for the name to mean 'Silver-tall' as a reference to Celeborn's tall stature.