What is a degenerate Fermi gas?

Degenerate usually means that the gas is in a quantum regime, that is the thermal de Broglie wave length $\lambda_{\rm dB}\propto T^{-1/2}$ is much larger that the distance between particles $l=n^{-1/d}$, where $n$ is the density and $d$ is the dimension of space. One then has $l\propto k_F^{-1}$ where $k_F$ is the Fermi momentum. This regime is the opposite of that of the classical (dilute) gas.

A degenerate Fermi gas is thus such that $k_B T\ll E_F=\frac{k_F^2}{2m}$. This corresponds to the limit where the fermions form a well defined Fermi sphere, etc described in text books (usually in a chapter about the Fermi gas). Note that electrons in metals also form a degenerate Fermi gas.