Energies of orbitals

The hydrogen atom: one electron, energy depends only on $n$. For hydrogen, since only a single electron is present, its energy in the $n$th orbit is fixed entirely by the formula $E_n=-\left(1312.8\,Z^2/n^2\right)$ kJ mol$^{-1}$ (with $Z=1$) - and notice that this expression contains no dependence on $l$ or $m$ at all. That means every orbital sharing the same $n$ has exactly the same energy in hydrogen, regardless of its subshell: the energy ordering runs

The 1s orbital, being lowest in energy, is the one the electron occupies in hydrogen's normal, unexcited state - the ground state. If the electron absorbs energy and jumps up to 2s, 2p or any higher orbital, the atom is said to be in an excited state.

Multi-electron atoms: the (n + l) rule. This tidy picture only holds because hydrogen has exactly one electron. In every other atom, the Schrödinger equation must account for electron-electron repulsion as well as nuclear attraction, which makes it far too complicated to solve exactly - so orbital energies in multi-electron atoms are instead ranked using an empirical guideline called the (n + l) rule: the lower the value of $(n+l)$ for a given orbital, the lower its energy; and if two orbitals happen to share the same $(n+l)$ value, the one with the smaller $n$ has the lower energy. Table 2.2 lists $(n+l)$ for every orbital from 1s up to 7s, and working through the table systematically in order of increasing $(n+l)$ (breaking ties by smaller $n$) reproduces the real filling order for multi-electron atoms:

Notice the famous case buried in this list: $4s$ (with $n+l=4$) comes out lower in energy than $3d$ (with $n+l=5$), even though $3d$ has the smaller principal quantum number - which is exactly why, as later sections show, the 4s orbital fills before the 3d orbital despite "3" being smaller than "4."

Degenerate orbitals. The three p orbitals of any given subshell - $p_x$, $p_y$ and $p_z$ - all have exactly the same energy as one another; orbitals that share identical energy like this are called degenerate. That degeneracy is a consequence of empty space having no preferred direction; switching on an external magnetic or electric field breaks that directional symmetry and lifts the degeneracy, splitting the previously identical energies apart (this is, again, the physical origin of the Zeeman and Stark effects).

Effective nuclear charge and the s < p < d < f ordering. Inside a multi-electron atom, an electron feels two competing electrostatic forces at once: the attractive pull of the positively charged nucleus, and the repulsive push of every other electron around it. The net pull actually experienced, after accounting for this partial cancellation, is called the effective nuclear charge. How much "shielding" an electron experiences from the other electrons depends on the shape of its own orbital, and it turns out that shielding grows steadily weaker (so effective nuclear charge grows steadily stronger) in the order $s>p>d>f$ within one given principal shell. Since a larger effective nuclear charge always means a more tightly bound, lower-energy, more stable orbital, this directly implies the energy ordering

$s<p<d<f\qquad\text{(within a given value of }n\text{)}$ …