These objects are the 'end stage' of the evolution of stars and represent objects supported against gravitational collapse, not by the ordinary thermal pressure of hot gases as in normal stars, but by something called degeneracy pressure. White dwarfs are supported by 'electron degeneracy pressure' while neutron stars are supported by 'neutron degeneracy pressure'. Electrons and neutrons are both fundamental particles that have 1/2 unit of quantum mechanical spin and are also called 'fermions'. We might recall from high school chemistry that the only way you can pack two electrons into the same atomic orbital is if one has its spin 'up' while the other has its spin 'down'. In white dwarfs and neutron stars, the same rule applies because the density of the electron and neutron 'gas' is so high that there are pairs of electrons and neutrons for which all the other quantum numbers are the same for the individuals of the pair. This leaves only their spin orientations as a remaining 'free' parameter, and it is the action of the Pauli Exclusion Principle that produces the degeneracy pressure.
Within the body of the white dwarf or neutron star, there are a finite number of permitted quantum cells distinguished by the particle's spin, energy and angular momentum, and once these cells are occupied by the electrons and neutrons that's it. Like a checker board in which all the squares are occupied by checkers, there is no room left for one more. This also means that for a finite number of electrons in a white dwarf, or neutrons in a neutron star, there is a well defined volume to the star in which the number of cells equals the number of electrons or neutrons. If I told you that I had 500 checkers, and that quantum mechanics dictated the size of the cell that each checker occupied, you could tell me how big the checker board could be!
Physicists have spent a lot of time trying to understand what is called the 'Equation of State' for degenerate matter. This equation describes just how much pressure a particular gas at a given density can produce, that would be available for supporting it against further gravitational collapse. This equation describes how the 'phase' of a gas changes at different densities, and once specified, you can slap it into a second equation that describes the equilibrium properties of the body under its own self-gravity.
At densities of 10^6 grams per cubic centimeter or so, the phase of the gas changes from a perfect gas, to one in which electron degeneracy pressure kicks in. The resulting equilibrium shape of the star is an object about 10,000 kilometers in diameter with masses less than the Chandrasekhar limit of 1.44 times the mass of the Sun. At higher masses, the object becomes unstable and collapses further.
At densities of about 10^14 grams/cc a second stable configuration appears in which neutrons, not electrons, provide the degeneracy pressure. The resulting equilibrium mass is the Oppenheimer-Volkov Limit of 1.4 solar masses, and the size of the 'neutron star' is about 10 kilometers in diameter. above this mass limit, the configuration again becomes unstable and collapses further.
Above densities of 10^14 grams/cc, physicists ar still undecided whether additional stable configurations exist. One long expected possibility is based on the fact that neutrons are not fundamental, but consist of three bound quarks. At densities above 10^14 gm/cc, it becomes more energetically favorable for the neutrons to dissolve as collective 'fermions' and give up their quarks which are also fermions and capable of producing their own degeneracy pressure.
A recent article by Erlend Ostgaard at the University of Trondheim in Norway, published in the journal Physics Reports in 1994, vol. 242, page 313 describes the current state of the art in figuring out what the equation of state looks like at densities above 10^14 gm/cc. There are several possibilities depending on just how you do the quantum mechanical calculations that describe how the strong interaction works according to the so-called Standard Model. If densities of about 10^15 gm/cc can be achieved in the core of a neutron star, quark matter will appear and you can end up with a hybrid star .From the outside, it looks like a neutron star, but the quark matter core allows the neutron star to be a stable object at about the same size of 10 - 13 kilometers as ordinary neutron stars, however, they would contain less mass: 1.4 - 1.7 solar masses compared to 1.8 - 2.6 solar masses because their quark-rich cores provide higher pressure than an equal volume of neutrons.
The structure of one such hybrid star show a core of quarks, actually 'strange quarks' out to a radius of 6-7 km radius occupying 60 percent of the star's mass, with an outer layer 3-4 kilometers thick of neutron matter.
In principle, you should be able to detect such quark stars because they will look like ordinary neutron stars in size, but have significantly lower total masses. This means that their moment of inertia will be lower and that they will spin faster as pulsars. They should be sub-millisecond pulsars. Out of 500 pulsars discovered, none have been found to spin faster than a millisecond, so this might suggest that, either quark stars are very rare and quickly evolve into, say, black holes, or that something is wrong with the theory and the equation of state being used.
For more information about neutron stars, visit University of Oregons Astronomy 122 Lecture