From Bulbapedia, the community-driven Pokémon encyclopedia.

A Wi-Fi League, or more specifically, a Pokémon Wi-Fi League is the colloquial term given to any internet-based, organized Pokémon gaming community that offers interactive game play via some gaming console and a Pokémon-related video game though online or Wi-Fi connectivity.

Lexical Origin

The nature of the colloquium is currently at argument among participants of Wi-Fi Leagues, and several arguments have been proposed.