[Legolas]

Elnimara is correct. Staying in Middle-earth into the Fourth Age would not automatically transform elves into mortal beings.

Elrond's half-elven family was given the choice - Elrond, his parents, and his brother. It was granted to them for Earendil and Elwing's heroic acts against Morgoth.

Because Elrond married a full elf, Celebrian, the choice was also extended to his children by her. Arwen and her brothers, Elrohir and Elladan, all had to make the decision (and they were supposed to do so before or shortly after Elrond left Middle-earth).

This choice had nothing to do with the One Ring. Peter Jackson may have made it sound so in the movie, but I think it was taken more literally than he intended. Many understood it to imply Arwen's life was magically tied to the Ring somehow. It's Elrond that says something like her fate is tied to it, isn't it? He means it in a logical/emotional way - not some binding, magical way. If the Ring is destroyed, she will stay with Aragorn. If the Ring is not destroyed, she has no reason to stay as Middle-earth will be overtaken by evil and Aragorn will die shortly.