The Pharaoh's Principles prove that you have attained a certain level of expertise, experience, or simple effort. With each Principle that you complete, you attain a new (character) level.
The first Principle that you must complete is that of Basic Citizenship.
Having passed that, you are eligible to work on other principles. With each principle, more and yet more other principles are available.
Each Discipline has an associated general Principle, which serves as an initiation into that discipline. To pass the initiation into that discipline, you must complete the Principle for that discipline.
Each Discipline also has 7 Tests. We currently believe that each Test has an associated Principle. As only a few Tests are currently known, time will tell. Where for Initiation, completing the Principle completes the initiation itself, completing the Principle for a test only represents partial completion of the Test itself.
Said another way, you can complete the Principle without having to complete the whole Test. Principles are generally a noncompetitive version of the associated test and are included in part to appeal to more casual players.
With the single exception of Basic Citizenship, each Principle must first be demonstrated before that Principle becomes available for citizens to work on. This demonstration takes the form of a task emblematic of the corresponding principle, or demonstrating a certain level of technical achievement, and is different for each Principle.
Where known, the demonstration task, who peformed it (and where), and when, is listed on the relevant page under Tests.