An invisibility sphere is a temporary item that the Mysterious Warrior gives you for the quest: Moonflower. It will make you invisible for 30 seconds. When you are invisible, enemies can't see you, thus they won't attack you. After you complete the quest, the Mysterious Warrior takes it back. It is stored in your Weapons section of your backpack. It is closely related to the Scrying Orb and Invisibility Bombs.