The "American Dream" as illustrated in The Great Gatsby is the idea that many American's extrapolate from the part of the Constitution which guarantees all citizens the rights of "...life, liberty, and the pursuit of happiness." The document only promises that Amerians will have the right to go after happiness. It doesn't promise that happiness will come.
However, many are disappointed when they don't achieve this happiness, because they operated under the premise that America was the land of opportunity (which they interpreted as a promise of the prosperity and success.)
It is this dream (that every American would have a loving family, success, and material wealth) that is explored in the Great Gatsby. Gatsby spends his life chasing after wealth so that he can get his love and his happy home. So, in a sense, he comes to symbolize all those who place their happiness on having these things.
When Gatsby is killed at the end of the book, I believe it is Fitzgerald's commentary that the American Dream also died. I think he meant to tell us that chasing after wealth and a finite definition of happiness doesn't bring happiness at all...only disappointment and heartache.