A dead star is a star that has exhausted all its fuel for nuclear fusion and is simply the core of the former star floating through space. The size of the star before it uses up all its fuel determines what happens to it afterward.
A star can either become a black dwarf, neutron star, or a black hole. Main sequence stars like the sun become a red giant by fusing helium. Then, the star ejects its outer layers, leaving an inner core of extremely dense matter supported by electron degeneracy pressure. This means that no two electrons can occupy the same position or energy state at the same time. The white dwarf glows with residual heat and eventually burns out into a black dwarf.
Stars greater than about three solar masses will go supernova and collapse into an even more compact form. Because nuclear fusion can no longer override gravity, the star's atoms collapse on themselves. Protons merge with electrons to form neutrons, which are then so tightly packed that a teaspoonful could weigh as much as a mountain.
The gravity of even more dense stars can overcome even neutron degeneracy pressure. Because no force after neutron degeneracy pressure can counteract gravity, the star collapses into a single point in space. Its escape velocity is so great that not even light can escape. The resulting black hole is a rip in spacetime that causes objects and information to be lost forever.