The battle box is a special box in your PC that you put pokemon in. In this box, you can put your 6 most preferred or best pokemon you use for battling your friends online or via C Gear (confirm?). It can also be used from the Battle Subway.
The battle box is also used for Official tournaments (like the VGC), where you connect with your opponent from the Main Menu, not giving you time in game to choose your pokemon, so you have to choose them and put them in your Battle Box before hand.
If you're EV training, or just training, and your friend asks you for a battle, you don't have waste time going back to your PC to deposit and withdraw pokemon. You connect with them, and then when they battle with you, the game will ask if you want to use the pokemon from your battle box.
Hope this helped.