It depends on how much mass the star has when it is born.
There are four different kind of ends of a star:
Very light stars Edit
The faintest stars, such as our nearest neighbor Proxima Centauri, just finish up all their hydrogen fuel and becomes fainter and fainter as they are cooling. They will simply go out. They will become what we call white dwarfs. A few of the more heavy of those stars will shortly become red giants before their fuel ends up and they faint.
Light stars Edit
Less light stars, like our Sun, will first become red giants, until their hydrogen will be used up. Then helium burning will start and they will become normal giants. At late age, when normal giants have used up the helium fuel, they will once again become red giants. This later kind of red giants becomes more and more unstable, they will kind of "hickup" by a sudden turn-on/turn-off of the helium burning. The last "hickups" will become more and more violent, until the star casts out its outer layers in a planetary nebula (either a ring nebula or a bipolar nebula). The star itself will shrink and become bluer and bluer, until it faints and becomes a hot white dwarf, that slowly will cool.
Heavy stars Edit
Heavy stars, like the star Rigel in Orion, will keep the same brightness all the time, but like all other stars it finishes up all its hydrogen in the core, and it becomes redder and redder, until the temperatures are high enough that it ignites helium, then it becomes again bluer. The hydrogen is still burning in a shell around the core. Step by step more and more fuels will ignite, the order after hydrogen and helium is carbon, neon, oxygen and silicon. This will go increasingly fast, and at the last stage iron is produced in the core, and the core will become more and more dense. At the end this heavy star will explode in a supernova explosion - the astronomers aren't sure on the details - which is about as bright as one galaxy when brightest. The star is disintegrated, but the core may still remain as a collapsed neutron star.
Very heavy stars Edit
Very heavy stars might simply collapse and disappear from sight. These heavy stars probably becomes black holes. The details are very insecure, and the astronomers believe yet more heavy stars explode in supernova explosion.