As it relates to The Great Gatsby, the American Dream has everything to do with the fact that our past does not have to dominate our future.
This is usually understood in the sense that a person born into poverty is not doomed to poverty. The class system in America is fluid. This is the essence of our collective, capitalist dream - you have the power to climb the economic ladder.
This novel takes the notion of self-creation to its logical limit with a character who not only climbs the class ladder and creates his own wealth, but he also fabricates an identity. Is this a perversion of the American Dream? Is it an extension of the American Dream?
Should we read the other characters, all highly concerned with class and wealth as these inform identity, as equally engaged in the process of self-creation?
I would say that part of the American Dream is the idea that anyone can do or achieve anything they want. Americans are not set into casts or other societal roles at birth. Anyone can be anything. Gatsby shows this part of the American Dream to be true. He also shows that dark side of this dream. Sometimes we find when we get where we wanted to go that it isn't what we thought it would be. Gatsby realizes that after everything he has done he still isn't happy nor is he where he thought he would be. He does rise financially but he this new financial gain does not provide him with the life he thought he would find.
I think much of this part of the American Dream remains the same. Anyone can become anything. The way in which children dream of becoming something more is really what has changed. Young people now talk about being famous or staring in a TV show like American Idol. There are no longer young people looking to the stock market or other areas of financial gain that existed in Gatsby's time. The idea is the same, but the manifestation of that dream has certainly changed.
The core of the American Dream is to the have the freedom to pursue health, wealth and happiness. This includes the idea that you can accomplish your goals through hard work and perseverance. I think that the American Dream is still this or some similar variation. Gatsby certainly embodies the dream in his childhood goals for bettering himself, his leaving school but learning about making money from people like Dan Cody, and his pursuit of wealth in the hopes of attaining future happiness with Daisy. Unfortunately, this novel shows the despair of the dream when it is not a reality.