The neutron star exists for the same reason the degenerate dwarf star exists: degeneracy pressure counteracts gravity, so that the star ceases to shrink as it cools. But the source of this pressure in a neutron star is not electrons, as it is in a degenerate dwarf star, but degenerate neutrons and protons. This simple difference is the source for the radical difference in size between the neutron star and the degenerate dwarf.
Neutrons and protons become at extremely high densities because this peculiar state is energetically more favorable than for them to remain in atomic nuclei. In effect, the atomic nuclei created over so many eons through thermonuclear fusion evaporate away under the high pressure in the core of a neutron star. As with a gas of free electrons, the protons and neutrons are in very specific energy states, with each energy state capable of holding only two particles of a given type. In a cold star, the neutrons and protons fully populate the lowest energy levels, with the density of neutrons nearly equal to the density of protons.
The name neutron star is not entirely accurate, because the complete neutron and proton degeneracy in a cold neutron star causes the density of neutrons and protons to be nearly identical. On Earth a free neutron decays into a proton and an electron, releasing a neutrino and a small amount of energy in the process. This cannot happen to a neutron in a cold neutron star because the proton would have nowhere to go; the proton must go into an energy state that at most contains only one proton, but no such state is available. Conversely, the inverse process, the absorption of an electron by a proton to create a neutron, cannot occur because all of the available neutron energy states are filled. This is why the number of protons and neutrons in a cold neutron star are nearly identical.
The relative size of the neutron star to the degenerate dwarf star is a direct reflection of a fundamental microscopic property of our universe. The degeneracy pressure depends on the mass of the particles providing the pressure. The radius of a star held up by degeneracy pressure is then inversely proportional to the mass of the particle providing the pressure. The precise relationship between mass and radius in a neutron star is unknown, as it also depends on how the protons and neutrons interact with each other, but the particle mass is the dominant factor in setting the radius. This is why neutron stars are so much smaller than degenerate dwarf stars; the proton's mass is 2000 times larger than the electron's, so a neutron star's radius is smaller than a degenerate dwarf's by roughly the same factor. So 15 kilometers is a typical value for a neutron star's radius, compared to the typical degenerate dwarf radius of 10,000 km.
A startling consequence of the small size of the neutron star is that its gravitation field is strong enough to exhibit some of the effects of general relativity. Two signature effects are the strong gravitational redshift of light leaving the star's atmosphere and the deflection of light passing by the star. These effects are strong because the radius of a neutron star is not many times larger than the radius of a black hole with the same mass. For instance, the event horizon of a 1.5 solar mass black hole is 4.5 km, where the radius is defined by the surface area of the event horizon. With a typical value of 15km, a neutron star's radius is around three times the event horizon radius. The gravitational redshift, which depends only on the ratio of the neutron star's radius to the event horizon radius of a comparable-mass black hole, is 16%. Because the light is deflected as it leaves the star's surface, we see more than half of the neutron star's surface. The fraction of the surface seen is again only depends on the ratio of the neutron star's radius to the event horizon radius, and therefore dependent only on the gravitational redshift.
One feature of black holes that should not appear around a neutron star is a last stable orbit. Around a black hole, one finds a minimum radius for the orbit of an object; inside this radius, an object must fall onto the event horizon of the black hole. This effect could only appear around a neutron star if the surface of the neutron star were inside this radius. This would require a stellar radius that is only 50% larger than the event horizon radius, which is a result not produced by most neutron star theories. If such an object existed, however, it should be easily demonstrated, because the gravitational redshift from a neutron star with a surface at the last stable redshift is a massive 42%.
The gravitation redshift is a direct measure of the ratio of the star's mass to its radius. For this reason, the measurement of redshift is an important measure of the structure of the star, and ultimately of the behavior of the material within the neutron star.