Doesn't it tell you the first time to enter one? I cant remember, i skipped the intro.
Eliminator is basically a battle royal. You start with access to the entire map but the playing area shrinks every few minutes, ending with a scramble to the finish line when there's only a few cars left. As the playing area shrinks, you have to find the best (highest numbered) car you can. you can also challange other players during that time and if you win you'll have the option to either take their car or upgrade yours by a level (from 1 to 10), but if you lose you're out.