How do orbitals fill, what is the difference between them, and why is this stable?
What is stable?

In this context, stability is that it neither gains electron to form negative anions nor loses electron to form positive cations. I will only discuss noble gases because they are so exceptionally stable.

It does not lose electrons

This is not unique to noble gases. Elements in group VII such as fluorine and chlorine also do not tend to lose electrons.

This is also not entirely true. Compounds like $\ce{XeF4}$ and $\ce{Cl2O7}$ do exist in which $\ce{Xe}$ and $\ce{Cl}$ bear partial positive charges respectively, although $\ce{Xe}$ is a noble gas.

$\ce{Xe}$, being a noble gas, indeed has a high ionization energy (energy taken to remove an electron) but it is not infinite. This means, as long as we have enough energy, we can make them form compounds.

The ionization energy increases from Group III to Group 0 because there are more protons in the nucleus which exerts a higher pull to the orbitals, which lowers their energy. For example, the ionization energy of helium is twice that of hydrogen, and helium has exactly twice the number of protons than hydrogen.

It does not gain electron

This is because the next orbital has a positive energy. Orbitals with negative energy means that it takes work to free the electrons from the orbitals. Orbitals with positive energy means that no electrons can reside in the orbitals: they are already free.

For example, the electronic arrangement of argon is $1s^2 2s^2 2p^6 3s^2 3p^6$. The next orbital, if it existed, would be $4s$, which has a positive energy. Therefore, its electron affinity (energy released by gaining electrons) is 0. This is evidenced by the ionization energy of the next element, potassium, which is small, suggesting that the extra proton pulls the $4s$ orbital just enough so that it has a slightly negative energy.

Conclusion

Noble gases are stable because they have a full $p$ orbital, so the next electron would need to go to the next orbital which has a positive energy, and the extra protons pulling on the orbital lower the energy of the electrons in the orbital, making it not tend to lose electrons.