Elves are immortal in a sense that their bodies never grow old, though their minds do age, and they cannot be stricken by illness. But, elves can be wounded to the point that their body is useless and they die, or they tire of the world and their spirits (fëa) choose to leave their bodies (hroa).

When elves die, their spirits are summoned to the Halls of Mandos, also known as the Halls of Waiting or House of Dead, where they await their fate. After some time, the elves’ spirits can choose to be re-born into a body exactly the same as the former one and be returned to Aman (where Valinor is located), or stay in the Halls of Mandos until the end of the World.

Elves that had done some evil in their lives before would be required to wait longer than others to be re-incarnated. However, elves that had done great evils in their past can not be permitted to return to Aman and must stay in the Halls of Mandos forever, such as the elf Fëanor.

It is possible that Glorfindel of Rivendell and Glorfindel of Gondolin are one in the same, and that he was re-born sometime during the second age and sent to Middle-earth. 

Written by Fánefalmë Raitallë (www.homeschoolblogger.com/jr90)

Sources: http://tolkien.cro.net/elves/phylosop.html The Grey Havens, Elvish Philosophy of Death and Incarnation, www.councilofelrond.com Encyclopedia search: Halls of Mandos, Glorfindel of Rivendell