Question

what are the three rules that govern the filling of atomic orbitals by electrons

Answer 1

three principles that govern this process:
1) the Aufbau (build-up) principle,
2) the Pauli exclusion principle,
3) Hund's rule

Exclusion principle
No more than one electron can have all four quantum numbers the same. What this translates to in terms of our pictures of orbitals is that each orbital can only hold two electrons, one "spin up" and one "spin down".

Build-up principle
You may consider an atom as being "built up" from a naked nucleus by gradually adding to it one electron after another, until all the electrons it will hold have been added. Much as one fills up a container with liquid from the bottom up, so also are the orbitals of an atom filled from the lowest energy orbitals to the highest energy orbitals.

However, the three p orbitals of a given shell all occur at the same energy level. So, how are they filled up? Is one of them filled full with the two electrons it can hold first? Or do each of the three orbitals receive one electron apiece before the any single orbital is double occupied. As it turns out, the latter situation occurs.

Hund's rule
This rule is applicable only for those elements that have d electrons, and so is less important in organic chemistry (though it is important in organometallic chemistry). It says that filled and half-filled shells tend to have additional stability. In some instances, then, for example, the 4s orbitals will be filled before the 3D orbitals.