the Fermi energy

The Fermi energy is the water line of the electron sea: the energy of the highest occupied electron level when a metal is cooled to absolute zero. Because electrons are fermions and refuse to pile into the same state, they must stack up one per level, so even at zero temperature the topmost electrons carry an enormous energy. That top energy, where the filling stops, is the Fermi energy.

Formally, E_F is the chemical potential of the electron gas at temperature T = 0, the dividing energy below which states are filled and above which they are empty. For a free-electron gas of density n, E_F = (hbar^2 / 2m)(3 pi^2 n)^(2/3); associated with it are the Fermi wavevector k_F, the Fermi velocity v_F, and the Fermi temperature T_F = E_F / k_B, which for ordinary metals is tens of thousands of kelvin. Because room temperature is far below T_F, the electron gas is strongly degenerate: the Fermi-Dirac occupation f(E) = 1/(exp[(E - mu)/k_B T] + 1) is a nearly perfect step at E_F, smeared only within a thin band of width about k_B T around it.

The Fermi energy sets the fundamental energy scale of a metal and governs everything from why the electronic heat capacity is small (only the thin sliver of electrons near E_F can absorb heat) to the degeneracy pressure that holds up a white dwarf star. A common looseness of language: at finite temperature the chemical potential mu(T) drifts slightly from its zero-temperature value, and the two are often both called the Fermi level; strictly, the Fermi energy is the T = 0 value.