Details:
To get to the Battle Tournament, you merely have to consult with the Aipom at the entrance to the main area of the Colosseum. This Aipom will invite you to join the tournament

The tournament has several rounds and has you face off against three Pokémon sequentially. You don't get health regeneration so you have to be careful with who you hit. You have the ability of changing Pokémon between the round of the tournament. The Pokémon you battle in this tournament are stronger than when you first faced them so you need to make sure your Pokémon are at a high level.

When you enter the mode, you have the ability to select two Backup Pokémon. Similar to how they'd come in a pinch during normal battles, these Pokémon will come in and attack the enemy, doing damage while they go. This is very useful in order to try and cover any type disadvantages brought on by the other Pokémon you face. To activate them, all you have to do is press the A Button at any time. This calls them into battle and they shall quickly attack, do the damage, and then leave. You can only use each backup Pokémon once per round, and you cannot skip one to get to the other one.