There are two versions called Bomberman Max: Blue Champion and Bomberman Max: Red Challenger. In Blue Champion the playable character is Bomberman where in Red Challenger the playable character is Max.
The story in both versions of the game is the same. An artificial intelligence named Brain has transformed five peaceful, happy planets into cold, dark, mechanical worlds. Both Bomberman and Max hear the distress cry of the Charaboms who live on these planets, and they begin a race to see who can save the Charaboms and defeat Brain first.
Bomberman and Max must conquer 100 different stages, all of which have the classic overhead Bomberman view. While there are various tasks on each level, gameplay mainly consists of good old-fashioned bombing.
A new on the game is the introduction of the Charaboms, which are similar to Pokémon in many respects. Each game has a unique set of Charaboms obtainable through single player mode.
Source: Wikipedia, "Bomberman Max", available under the CC-BY-SA License.