What is a degenerate Fermi gas? In ultracold atoms, it is generally talked of a degenerate Fermi gas. What does degenerate mean here?
 A: 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.
A: Degeneracy here refers to the property that most/all states with energy less than the Fermi energy are filled in a degenerate Fermi Gas.  Because of the Pauli Exclusion Principle that means that if you add an electron it cannot occupy a state that is already occupied. 
A: Agree with previous answers and I would like to add some part. 
When we talk about fermi gas, we know it's different than ideal gas even we neglect the interaction between particles. The quantum feature exhibits itself in two way: quantizing feature and statistical feature. For energy gap of the system $\Delta E\ll k_BT$, the physical quantity can be considered as quasi-continuous and quantizing feature could be neglected (which is the normal condition), but the statistical feature could still remain, and the condition is called degenerate condition. That means, whenever the statistical feature influences the system's property (partition function, say) dramatically, we call it degenerate. 
In physical interpretation, it means the condition always requiring a large overlapping between different particle's wave function. 
