Cherrygrove Pokémon Center from Pokémon Crystal.
Pokémon Centers have existed since the first game and are used to heal your Pokémon. They are also an access to PCs for storage of Items and Pokémon. As the games progress you get access to more including trading and battling other's Pokémon via a link. The first center is in the next city or town from your original city or town. You can also heal at the lab in your home city or town. When you fly or teleport it will be to the Pokémon Center if your going to a city or town (there are exceptions you can fly to in some games).