Degenerate matter exhibits quantum mechanical properties when a fermion system temperature approaches absolute zero. These properties result from a combination of the Pauli exclusion principle and quantum confinement.  The Pauli principle allows only one fermion in each quantum state and the confinement ensures that energy of these states increases as they are filled. The lowest states fill up and fermions are forced to occupy high energy states even at low temperature. 
While the Pauli principle and Fermi-Dirac distribution applies to all matter, the interesting cases for degenerate matter involve systems of many fermions. These cases can be understood with the help of the Fermi gas model.  Examples include electrons in metals and in white dwarf stars and neutrons in neutron stars. The electrons are confined   by Coulomb attraction to positive ion cores; the neutrons are confined by gravitation attraction.  The fermions, forced in to higher levels by the Pauli principle, exert pressure preventing further compression.
The allocation or distribution of fermions into quantum states ranked by energy is called the Fermi-Dirac distribution. Degenerate matter exhibits the results of Fermi-Dirac distribution.