quantum numbers
Quantum numbers are the small set of integers (and half-integers) that name a quantum state -- the atom's address system. Because bound quantum systems have discrete, quantized properties, you do not need continuous coordinates to specify a state; a handful of labels suffices. For an electron in an atom, four quantum numbers pin down its state completely, and no two electrons in an atom may share all four (the Pauli principle).
For the hydrogen atom the four are: the principal quantum number n = 1, 2, 3, ..., which sets the energy E_n = -13.6 eV/n^2 and roughly the size of the orbital; the orbital (azimuthal) quantum number l = 0, 1, ..., n-1, which fixes the orbital angular momentum magnitude sqrt(l(l+1)) hbar and the shell letter (s, p, d, f); the magnetic quantum number m_l = -l, ..., +l, which fixes the projection L_z = m_l hbar and thus the orbital's orientation; and the spin magnetic quantum number m_s = +1/2 or -1/2. Each of these labels a specific conserved (or, in an isolated atom, sharp) quantity, and they are exactly the eigenvalue labels of a maximal set of commuting operators -- H, L^2, L_z, and S_z -- which is the precise reason four of them, and only four, are 'good' at once.
The deeper point is that a quantum number is only 'good' when its operator commutes with the Hamiltonian, so which labels are valid depends on the physics. Turn on a spin-orbit interaction and m_l, m_s stop being conserved separately; the good labels become n, j, m_j (the total angular momentum), because now only the total is conserved. Quantum numbers are not decorations -- they are the conserved quantities of the problem, and choosing the right set is how you make a hard Hamiltonian tractable.
The four quantum numbers of a ground-state hydrogen electron are n=1, l=0, m_l=0, m_s=+1/2 (or -1/2) -- the 1s state. A second electron added to make helium must, by Pauli, take the other spin (m_s=-1/2), which is why the 1s shell holds exactly two electrons and the periodic table begins the way it does.
Four quantum numbers name each electron state, and Pauli forbids any two electrons from matching all four.
A quantum number is 'good' only if its operator commutes with the Hamiltonian; m_l and m_s are good for a bare Coulomb atom but not once spin-orbit coupling matters, where n, j, m_j take over. Do not treat the four hydrogen labels as universal -- the correct set is dictated by which symmetries the Hamiltonian actually has.