In a many-electron atom, electrons occupy one-particle states characterized approximately by orbital and spin quantum numbers. The Pauli exclusion principle allows at most one electron in each complete one-electron quantum state.

For a given orbital angular-momentum quantum number $\ell$, there are $2\ell+1$ values of $m_\ell$. Each orbital state can combine with two spin projections, so a subshell can contain $$2(2\ell+1)$$ electrons. Thus an $s$ subshell ($\ell=0$) holds $2$, a $p$ subshell ($\ell=1$) holds $6$, and a $d$ subshell ($\ell=2$) holds $10$.

The hydrogen atom's exact $1/n^2$ degeneracy is broken in many-electron atoms by electron-electron repulsion, shielding, penetration and relativistic/spin-orbit effects. Electrons fill available low-energy orbitals subject to antisymmetry, producing the familiar shell structure.

For example, carbon has six electrons. In an independent-particle picture its configuration is $$1s^2,2s^2,2p^2.$$ The two $1s$ states and two $2s$ states are filled with opposite spins, while the remaining electrons occupy states in the $2p$ subshell.

Electron configurations are therefore not arbitrary bookkeeping rules. They arise from quantized bound states, angular-momentum degeneracy, electron spin, exclusion, and interactions among electrons. This physical structure underlies valence, periodic chemical trends, atomic magnetism and the formation of solids.