Depending on it's size, many things can happen.
Smaller stars can simply burn out and become brown dwarf stars. The largest of stars collapse on themselves in what is called a supernova and can even become black holes.
Middle size stars, such as our sun, will swell up in size as they begin fusing heavier elements. All stars are powered by fusing elements into heavier elements such as fusing hydrogen into helium. Once it begins fusing the heavier elements the star swells in size. Our sun will become a red giant and swell to a size that will swallow the earth (fortunately for us that's a few billion years away from now).
It will burn as much fuel as it's size will allow then collapse in a far less spectacular way compared to a supernova and likely become a white dwarf. This smaller, hotter star will continue to burn until it runs out of steam.
There are so many different types of stars and so many different paths they can take from birth to death, that this website will offer you far more info than can be posted here:
http://www.duke.edu/~teb/stars/index.html