From Bulbapedia, the community-driven Pokémon encyclopedia.
The Pokémon Fan Club is an organization spread throughout the Pokémon world, with at least one Club House in nearly every region. The Fan Club is basically a place for people to meet and chat about their favorite Pokémon and to show off their collections to eachother. In general, the members are usually more oriented to unevolved, cute Pokémon and Pokémon Contests than Pokémon Battles.
The President of the Pokémon Fan Club has thus far appeared in every game. Gabby and Ty often visit the Fan Club House in Hoenn to do interviews with enthusiastic trainers.
Trainers of the PokéFan trainer class are members of the Pokémon Fan Club.
In the Pokémon Specia manga
The Pokémon Fan Cub is introduced in Danger: High Voltorb. Is a Pokémon lover association whose headquarters are located in Vermilion City, this building's distinctive feature is a Pikachu logo in the front of it. Inside, a group of middle-aged PokéFans talk about their favourite Pokémon and read their own magazine. They hate to force Pokémon into battles and they prefer un-evolved ones, because they're cuter. The head of the society is the odd Pokémon Fan Club President, whose Abra was stolen.