Inheritance is a state of being safe from any loss of bank items (gold and ender chest contents) when dying. This is a one time working state which must be replaced after death. If one dies without inheritance, you loose all your bank contents upon Reincarnation.
Inheritance can be bought from Bankers by clicking on the Thumbs Down button. Once purchased, this Thumb wil face up. Inheritance costs 500 gold.
You can also check if you have purchased inheritance through your profile card in the Player Menu.