Surprised no one asked this but it is what it says.
The battles themselves or the whole process? The battles take place on our Showdown server. There isn't really anything special about them, both players simply agree to follow the rules and only use Pokémon that are allowed.
As for the process itself, it goes as follows:
Here's suwmun's guide on how to host a tournament
And an example of a past tournament thread.
If any of this is incorrect please let me know and I'll edit it.
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