Elrond always said she would lose her immortality if she stayed, and Aragorn was one of those humans that lived a really long time (like centuries) so how was this an issue? Eh, again, my memory's probably rusty.
My unfettered nerdiness made me answer:
It was an issue because elves only have eternal life on Arda (the world). They can be reborn under certain circumstances (if I remember correctly), but they are tied to the world, so when they die, it is a fairly deep tragedy. I would imagine that Arwen losing her immortality might also mean a true, complete and permanent death for her, but I'm not certain. Aragorn is human. They live shorter lives in the world, but they return to Eru (the supreme deity in LotR) when they die, and live there forever. So Arwen's sacrifice is a rather serious thing, and Aragorn might be the most selfish asshat ever for letting her go through with it (although I do appreciate that she actually makes an autonomous decision in this). He already gets his eternal life after his body dies. Though it makes for a rather pleasantly poignant love story. I only wish that Arwen got to have an actual personality (beyond being soooooo pretty
) to go with it.
Been a long time since I read these books, though. I may be misremembering things.
However, that does make me wonder how they will handle the immortality issue with Solas and Lavellan, assuming that there is even a scenario where they will be able to continue their relationship.
... what? I'm totes on topic. Shut up. 