Electron degeneracy occurs when matter is compressed to the point where electrons fill the lowest quantum states, which the Pauli exclusion principle says may each be filled with at most one electron, given a sufficiently small volume. The result is a pressure (electron degeneracy pressure) against further compression. The word degeneracy is used to indicate that at this compression, ideal gas laws no longer hold, i.e., temperature does not affect it. It provides the force that keeps a white dwarf (or some gas giants) from collapsing into a neutron star.
Electron degenerate matter and electron degeneracy pressure are instances of degenerate matter and degenerate pressure. Another is neutron degenerate matter and neutron degenerate pressure analogous for neutron quantum states, and describes the material of neutron stars, and possibly the center of some white dwarfs.
With sufficient gravity, these pressures can be overcome, producing a black hole. Though it isn't clear what the microscopic conditions are in the extreme conditions behind the event horizon and/or at the singularity, it is assumed the matter is forcibly transformed to bosons (e.g., photons), which are not ruled by the Pauli exclusion principle.
Given electron degeneracy, the Fermi energy is the energy level of the highest quantum state occupied if the material is at absolute zero.