Neutron stars are stars, like the degenerate dwarfs, that no longer generate energy through gravitational contraction and core thermonuclear fusion. But while the degenerate dwarf remains large on human terms—about the size of Earth— the neutron star is small enough, with a typical radius of 15km, to fit inside a city. With this small size comes a density that rivals the density of an atomic nucleus, because while neutron stars are tiny, they are also more massive than the Sun. A neutron star is not much larger than the event horizon of a black hole of the same mass, so the effects of general relativity are strong in around a neutron star.
A neutron star is the metamorphose of a large star after core collapse. Once a star larger than about 10 solar masses consumes all of its thermonuclear fuel, core collapse is inevitable. Without nuclear fuel to generate heat, the star is unable to halt gravitational collapse until a new source of pressure are tapped that can counteract the gravitational force. Degenerate protons and neutrons provide this source of pressure. The resulting halt to the implosion releases energy that drives a supernova, and emerging from the center of the supernova remnant is a new neutron star.
The small surface area of a neutron star makes it difficult to see. In fact, only several isolated neutron stars have been seen by the light they emit from their surfaces. We see many neutron stars, but for reasons other than their thermal radiation. We see isolated neutron stars that convert their rotational energy into radio waves, and we see neutron stars in binary systems that convert gravitational potential energy into x-rays and gamma-rays.
The isolated neutron stars that we see are called radio pulsars. These stars have incredibly strong magnetic fields, and as the star rotates, the oscillating magnetic field drives outward an electromagnetic wave, most of which we usually see as radio waves. The energy in these radio waves comes at the expense of the star's rotational energy, so over time the radio pulsar's rotation slows. Radio pulsars can also be sources of pulsed visible and x-ray light.
We see neutron stars in binary systems when they pull gas from companion stars onto themselves, causing the gas to generate x-rays. Naturally enough, these systems are called x-ray binaries. If a neutron star in an x-ray binary has a strong magnetic field, the x-ray emission is modulated by the spin of the star; these binaries are called x-ray pulsars. The amount of energy released when gas falls onto the surface of a neutron star is a massive fraction of the total rest mass energy of the gas. For instance, if an object free-falls to the surface of a neutron star, where it is brought to rest, and all of the gravitational potential energy is released into space as light, an observer would find that 15% of the object's rest mass energy is converted into radiation if the star redshifts light by 15%. This is why neutrons stars in compact binary systems release so much energy.
These two varieties of neutron star system are very different in their power generation. The radio pulsars generate small amounts of power, so the radio pulsars we see are nearby. X-ray binaries, on the other hand, generate tremendous amounts of power, making them among the most luminous objects in the Galaxy. They dominate the x-ray sky, but as with the brightest visible stars, they are very far from us. In fact, many of the x-ray binaries we see are on the opposite side of the Galaxy.