During a supernova explosion, most, but not all, of the star is destroyed. A remnant will remain in which even the strength of electron degeneracy wont be enough to hold up against the pull of gravity. The electrons in this object will literally be crushed out of their lowest orbitals into the nucleus of the atom. When this happens, the electrons combine with the protons to make neutrons, and it will be as if the star is composed of one great big atomic nucleus, consisting entirely of neutrons. In contrast to a white dwarf, which is a solar-mass star crushed down to the size of the planet earth, this neutron star is a star crushed down into the size of State College.
Neutron stars have an interesting property, which involves their rotation rate.
Consider an ice skater, spinning slowly with her arms extended. This is due to conservation of angular momentum. Stars also conserve angular momentum. This is what a neutron star does. Because neutron stars are so small, their total thermal emission is negligible.