PokeBank is a transferring system to transfer Pokemon from previous games to the new followed up generations/created games.
To use PokeBank, you must install it from "Nintendo E-Shop" along with "PokeTransporter", They are both needed; plus they're free.
Once done, you need to have a copy of either:
>- Pokemon Black/White 2 with Pokemon X/Y
- Pokemon X/Y with Pokemon OR/AS
It works like this:
1) Log onto your primary game (the one with the Pokemon you're transferring).
2) Go into your PC boxes and put all your transferring Pokemon into BOX 1
3) From that box, hit home menu and close off your game.
4) Go onto PokeTransfer, and it will transfer the box to Pokemon bank (this can not previously returned once transferred).
5) Switch to you secondary game (the game you're giving the Pokemon too)
6) Once the game is in, log onto PokeBank, and you'll be able to put your Pokemon into your boxes, in your secondary game. At this point you can store and receive Pokemon with your secondary game.
7) Transfer Complete!
You can store up to 3,000 Pokemon hacked/not - basically a much bigger PC when the ingame one fails. At first, pokemiles and BP can be received; along with a free lvl. 10 Celebi. It's a really good experience and definitely recommend having it.
Although this system is fantastic, only the first month that you've downloaded this system will be considered free. After expired, you would have to pay $5 (real money) for a year after - for Canada
Source/More Info: Prices, Using PokeBank, Using PokeTransporter, and Own Experience.
Hope I helped! :)