So my parents and I got into a fight which resulted in them taking away my Xbox for good. Any ideas on how I could get them to change their minds so that I might get it back?

Depends. How pissed are they and why?

I guess this can depend on what the fight was about and how serious it was.
One way, if they believe that the fight resulted due to something you did, is to apologize. Not just a "I'm sorry", but a true apology.
Another way is wait for about a day or two without your Xbox and then coming up to them that you've finally learned your lesson (for whatever it is) and won't try to do anything like that ever again.
Try to bargain with them, I guess? Convince them to give back the Xbox.
There are many ways you can get about this. Just remember to leave your pride out of this and just deal with whatever nagging comes you way.

Try and earn it back. If they just have it sitting on a shelf or something it means that you can probably get it back. Try doing extra work to convince them to give it back.
if that dosent work you can eithermwait until you move out of the house to get it back or you can buy a new one. (I have a friend whose last name is nelson, has an xbox,and fights with his parents a lot so it was wierd seeing this post. It makes wonder if thats you.)