First of all, welcome to the Reading Room! And no spoiler tags are
necessary here. :)
As descendents of Earendil and Elwing, Elrond Halfelven and his children were
given the choice to be counted among the Elves, and thus be immortal (Elrond’s
choice), or to be counted among Men and become mortal (the choice of Arwen and
Elrond’s brother Elros). Even though they had long lives, they eventually
died and their souls left the circles of the world. Arwen made the choice
she did so she wouldn’t be separated from Aragorn forever (or until the end of
the world). After Aragorn's death, she lost all joy in living and gave up
her life soon after in Lorien.
Luthien made a similar choice so she wouldn’t be separated from Beren, but it
wasn’t because she was Halfelven. She went to Mandos after Beren died and
begged him to let her be reunited with Beren. With the permission of
Iluvatar, Mandos granted her wish and the result was that she became mortal and
died (after being granted a brief time back on earth with Beren).
FYI, the Tale of Aragorn and Arwen can be found in the Appendices at the end of
ROTK. It explains a lot of this and is a great (if short) story. I
highly recommend reading it. The tale of Beren and Luthien can be found
in The Silmarillion and is also a wonderful read.
Finally, I put a link to TORN 101 below. It's a great little site packed
with lots of tips on posting here. Enjoy and welcome
again!