The Registry is basically various sets of Stickers that one must collect in order to get rid of the Calamity. It is composed of various categories (more like skills) that you will improve with time.
Collecting and completing Foundables will increase your statistics in the category (eventually levelling up your Rank in a given Registry) that the Foundable belongs to. Levelling up a Registry/Rank will reward you with Scrolls. Below are the various categories from the Registry, along with the various Foundables they each contain.
Infographic from Orange Wizard 2019