After a red giant has finished burning through the remaining helium it moves on to the next stage of existence. Like the amount of time spent in main sequence, what happens next is depends on the mass of a star.

Small stars become white dwarfs. These stars burn carbon, but are never hot enough to cause nuclear fusion. Eventually white dwarfs cool to become black dwarfs. The end result is a star that never collapses, but eventually runs out of head and energy, becoming a lifeless mass of matter.

Mid-sized stars become neutron stars. Neutron stars are similar to white dwarfs, but with a few important differences. One is that a neutron star contains neutrons at its core. Another difference is that neutron stars have supernovas. A supernova is a final nuclear explosion that generates enormous amounts of heat and light. Some of the best pictures astronomers have taken of the universe are these spectacular explosions.

Large stars will also become neutron stars over time. However, there is another possibility with large stars: the chance to become a black hole. Some stars are so large that when they collapse the matter begins to focus on a single point. Through a process that scientists are still trying to understand the matter becomes so extremely dense that gravity forms a funnel that traps all kinds of matter into it – even light itself. Black holes are a fascinating part of astronomy and scientists are only beginning to understand them.

Sources
http://aspire.cosmic-ray.org/labs/star_life/starlife_end.html (fair use)