Your inventory stores all of your items that may be needed for quests or to construct certain buildings. You can access the inventory by tapping the first button in the lower right corner of the screen.
Items are split into categories. Tap a category to see what items you have. A gray category has no items in it.
You can use some items like Energy Refills from your inventory. Tap this item once to utilize it.
Note, however, that it's not possible to use quest items directly from your inventory, because they are either used automatically or require you to tap "Use" in a corresponding quest window in your To-do List.
See the article "Completing quests" for more information on using items in different quests.