Pokémon Bank is a piece of downloadable software that allows for you to store various Pokémon from your Pokémon X & Y games on the internet. It requires a yearly fee of 500¥ and has storage for up to 3,000 Pokémon. It also runs a trial for 30 days at any point until January 31st which is when it becomes a paid service.
Pokémon Bank simply allows you to store up to 3,000 Pokémon within 100 boxes on the Internet, away from the boxes you have in your game. With this, you also have the ability to transfer content between various Pokémon X & Y games. It also has protections that prevent cheated Pokémon from being able to be transferred to the Bank.
In addition to that, you have the ability to search through your boxes based on type, species and various other features.
Each download of Pokémon Bank comes with the special app called Poké Transporter which allows for the transfer of Pokémon from Generation V games to Generation VI