The Inventory was introduced in v3.4 that allowed you to go to the Pokemon Center. You can access it by going to the main menu and pressing the inventory button. All obtained items and Avatars can be viewed in the inventory, and here you can change your Avatar. The inventory has nothing to do with using Evolution Stones, however.
NOTE: Evolution stones can only be used in game, not in inventory! To use them, go to the "Check Pokemon" page, and select "Use Item" to evolve it.