The Inventory is used to store items. Your Inventory is opened by pressing the Y BUTTON (default) and shows items not in your Hotbar but items that are equipped with you and can be used when dragged into the Hotbar.
In Survival mode, press the Y button to open up your inventory, to drag items into your hotbar and drag items into your Inventory.
If you die you loose everything in your inventory, unless you select 'keep inventory' at the start menu. Doing this will mean you can no longer get achievements on that world however.
The Inventory is replaced by a list of all the Blocks and Items. The list of items is brought up with the X button, while the Inventory can still be opened by pressing the same Y button.