OK, just to get this out of the way, Pokemon Bank and PokeTransporter have not been released yet. They were scheduled for release on December 27, but due to sudden unexpected amounts of traffic on the eShop servers in Japan (where the games were released earlier), the eShop crashed. When it came back up, Pokemon Bank and Transporter were not available for download, and Nintendo have said nothing about when we'll be getting the apps. However, we still have a wealth of information on how the apps work.
Basically, you need to go on the Nintendo eShop and download Pokemon Bank (and PokeTransporter is you want it). You will need a Nintendo Network ID to do this however. This is easy to set up, and it shouldn't take more than a minute.
Once you have successfully downloaded Pokemon Bank, insert Pokemon X or Pokemon Y into the game card slot and open Pokemon Bank. Brigette, an in-game character, will introduce you to the app, and will give you a brief run through of how things work and what button do and such.
Anything specific from here in unknown since the apps aren't around for most of us, but we do know that Pokemon a Bank will allow you to take Pokemon from you PC Boxes in Pokemon X or Y and store them online, freeing up space. You can store 3,000 Pokemon online, so don't worry about running out of space. You can take back Pokemon you have stored online, search for them, etc.
However Pokemon Bank is not free. Depending on your region, an annual fee of roughly 5 bucks must be paid. Not much of a worry really.
If you are still confused, check out Pokemon Bank's website. Nintendo probably explains it better than I do, and they will also explain how PokeTransporter works and how you can get a special Celebi that knows a rare move.