When a star leaves main sequence for good, it eventually dies and leaves behind a remnant. In the case of a small star, it simply fades out and becomes a white dwarf. For a medium star, roughly 4 Sol masses or less, there is red giant phase, followed by a fairly tame explosion as the core collapses into a white dwarf and ejects a planetary nebula. In the case of a larger star, a red supergiant is formed, which eventually explodes in a Type-II Supernova, and the resulting core collapses further into a neutron star. If the star is truly massive, larger than somewhere around 20 to 50 Sol masses (depending on who you ask), after the supernovas it will collapse into a black hole.
A white dwarf is a stellar corpse composed of a packed degenerate gas. Although it no longer undergoes nuclear fusion, it cannot collapse any further due to electron degeneracy pressure (via the Pauli Exclusion Principle); it will eventually cool -- over a period of one trillion years -- to become a black dwarf. It is the densest form of matter possible where individual nuclei still exist. White dwarfs tend to have an Earth-size radius.
A neutron star is a stellar corpse composed entirely of neutrons. If a white dwarf would exceed the Chandrasekhar Limit (about 1.4 Sol masses), electron degeneracy pressure fails. The protons of the nucleus merge with the electrons in a massive shower of neutrons and electron neutrinos. At that point, Pauli takes over again and neutron degeneracy pressure holds the star in place. Essentially, the star becomes a single, giant nucleus. Neutron stars tend to be about 10km or so in radius.
A black hole is the result of a stellar core too large for even neutron degeneracy to support. If the resulting neutron star would be larger than about 3 Sol masses, the star collapses beyond its Schwarzchild radius and becomes a black hole. Nothing can escape from a black hole (except Hawking radiation).