What is the difference between a white dwarf, a neutron star and a black hole?

White dwarfs, neutron stars and black holes are all possible final stages of evolution of stars. In a star, a large amount of matter is collapsing together due to the gravitational attraction. However, as matter particles get close enough together, they undergo nuclear fusion which releases energy (in the form of radiation) that stops the star from collapsing. As the result of the fusion, lighter elements (such as Hydrogen) bond together to form heavier elements, but when the heavier elements undergo fusion, less energy is released. So, eventually the star runs out of fuel. What happens then depends on the mass of the star.

If the mass of the star is less than about one and a half  masses of the Sun, it becomes white dwarf. White dwarf eventually stops collapsing because of the pressure provided by electron degeneracy. This is a quantum mechanical principle that roughly could be interpreted as that the two electrons cannot be in the same place at the same time, even if the forces of gravity are pushing them together. Because white dwarfs have (relatively) small mass, this pressure is sufficient to stop them from collapsing further. The Sun is eventually going to become a white dwarf.

For more massive stars, electron degeneracy is not sufficient to stop the collapse, and the process continues until the star is composed mostly of neutrons (a neutron is produced when a proton and an electron are combined.) Then, for the stars with the mass less than about two or three masses of the Sun, the gravitational collapsed is stopped by the neutron degeneracy. The resultant product is called a neutron star. Its density is billion time greater than the density of a white dwarf.

If the mass of the star is greater than the three masses of the Sun, its gravitational collapse cannot be stopped by the neutron degeneracy. This kind of star eventually becomes a black hole - a large amount of mass condensed in a small point in space, an extremely dense object. It is called a black hole because once the star becomes smaller than a certain size, its gravity is so great that nothing - including light - can escape it. This means we can gather no information about what happens within that region of space.

Please see the reference links for further reading.