Quantum degeneracy pressure is closely related to a separate concept, the Pauli Exclusion Principle, which is responsible for a great number of phenomena that we observe in chemistry and physics, include the effects of quantum degeneracy pressure.
The Pauli Exclusion Principle governs some of the behaviors of a certain type of matter, called a fermion. Fermions include all of the matter particles that we are familiar with, such as protons, neutrons, electrons, and most atoms. The PEP states that two of these particles cannot occupy the same place at the same time (specifically, it says that two particles with the same quantum spin can't occupy the same quantum state, which is a more mathematical definition that basically says the same thing).
Electrons, in particular, are strongly influenced by the PEP, because this prevents them from cramming together as close to the nucleus as possible. This leads to electron orbitals, which leads to chemistry. It also leads to the commonplace perception that matter is impenetrable; this is normally ascribed to the electron clouds of atoms electromagnetically repelling each other, but it has as much to do with the inability of electrons to be crammed into a smaller space without an enormous amount of energy to do so. Electron degeneracy pressure is the pressure exerted "out" of the atom that prevents the electrons from being shoved downward, or "degenerating" from their current locations, into what are presumably filled locations.
EDP can be overcome with enormous energies, such as in a collapsing star. A star of sufficient mass (in theory, 1.4 solar masses or more) will be able to overcome EDP and force its electrons to collapse. Under these conditions, it is energetically more favorable for the electrons to combine with protons, converting them into neutrons. The resulting object is a neutron star, which is composed almost entirely of neutrons and does not actually perform stellar fusion. The neutron star is protected from further collapse by neutron degeneracy pressure, which is essentially the same governing idea as EDP. A star of even greater mass will be able to overcome even the NDP and become a black hole.