A phoenix is an eternal creature that possesses a body which can reincarnate itself on death, while maintaining the same DNA structure and memories.
It's a physical manifestation of the life stream, the endless mana flow of Eternia, with the two halves symbolizing opposites--the red phoenix, birth, and the black phoenix, death.
The red phoenix has flames that will never extinguish, whereas the black phoenix has flames that will erase anything it touches. As such, the two entities are unable to harm each other. They wage war by using human avatars.