The humans became ringwraithes, because they were easily corruptible, I think.
The elves didn't because they made their own rings. Or at least I'm assuming that's the reason they didn't. As far as I know, Sauron actually never came into contact with the elve's rings so he could never corrupt them.
Why didn't the dwarves become ringwraithes?
Is it because dwarves are more resistant to magic then the other races? That's the case in the Warhammer universe, I'm not sure if it's the same in Middle Earth.