Electron degeneracy occurs when matter is compressed to the point that electrons fill the lowest quantum states. The Pauli exclusion principle dictates that within a sufficiently small volume, at most one electron can be at any specific state. The result is a pressure (electron degeneracy pressure) against further compression. Another way to look at it (that explains more to me) is that the Heisenberg uncertainty principle demands that if the electrons are further confined, then their momentum, and thus speed must rise, which requires energy, a requirement enforced by a resistance to compression that must be overcome: the energy to make progress against this resistance provides the energy for accelerating the electrons, thus the pressure. During compression, as electrons take on momentum, they gain too much to maintain a nucleus orbit, and they travel freely through the material rather than remain with a single nucleus.
The word degeneracy is used to indicate that at this compression, the ideal gas law no longer holds: specifically, the temperature no longer has the same relation to pressure and volume. The pressure provides the force that keeps a white dwarf (or the stellar core of some gas giants) from collapsing further, e.g., into a neutron star. An analogous kind of degenerate matter and degeneracy pressure. is neutron degenerate matter and neutron degeneracy 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 underneath the event horizon and/or approaching 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.