From Bulbapedia, the community-driven Pokémon encyclopedia.
A Pokémon champion or League champion (Japanese: リーグ優勝者 League champion or simply チャンピオン Champion) is the highest known level of rank for a Pokémon trainer. It usually refers to a trainer who has completed a Pokémon League by collecting all the necessary badges, and either winning a regional Pokémon tournament or defeating the Elite Four and the previous Pokémon champion. There is usually only one Pokémon champion per region at a time. It is assumed that the champion must defend their title against challengers, although this is never explicitly stated.
In the games, Kanto and Johto share a League champion, while the other regions with a Pokémon league have their own.
Some Pokémon champions:
Strangely, it seems that the games' battle against the Elite Four and current Champion is not present in the anime, and likewise the anime's general tournament is not present in the games. However, this may change the season after Diamond & Pearl.