Pokémon-Amie is a new feature in the games Pokémon X and Y. Pokémon-Amie means "Pokémon-Friend" in French.
In Pokémon-Amie, you can care for your Pokémon and you might be able to raise the bonds and friendships between you and your Pokémon. If you make them happy, then they might look at you with a smile in battle.
There are different ways to care for your Pokémon.
There are different kinds of treats that you can feed to Pokémon. They might become friendlier or become angrier depending on the treats you feed the Pokémon.
Use the stylus to pet your Pokémon. The friendship can be raised depending on where the Pokémon likes to be petted.
There are ways to play with your Pokémon. The only way we know is that you can mimic your Pokémon's actions. (Examples: Open mouth wide, medium smile, tilt head, wide eyes).