Kennou. Authorities are special abilities that fall into a different category than Codecasts, Skills, and Noble Phantasms.
An Authority is a power that is on the level of creating a world, and includes things like altering events, time-flow manipulation, and kingdom building.
Authorities existed in the age known as the Age of Gods, which was about 6000 years ago, but after entering the Common Era human civilization advanced to the point where Authorities were no longer needed, and so Authorities became a relic of the past.
A normal skill “is able to accomplish a certain task by following a corresponding principle,” but an Authority works “simply by making things happen because one has that right.”
A divine spirit Servant should naturally possess Authorities, but to use them in the modern age requires a corresponding compensation to be paid (self-destruction).
In CCC the one who swallowed up Moon Cell has reached a level of power that is Authority class.

Fate/EXTRA Material: Encyclopedia of Fate/EXTRA