In this novel, the American dream is presented as the idea that anybody, no matter what their socio-economic background, can achieve material and social success. This is best shown through the character of Gatsby, who is able to overcome his humble origins and reinvent himself. By the time that Nick meets him, Gatsby is one of the wealthiest and most successful people in West Egg. Moreover, his lavish parties attract people from all over New York.
As the reader gets to know Gatsby, we also find that his American dream is tied up with a desire "to recapture the past"—specifically, to be reunited with his pre-war sweetheart, Daisy. Remember that Gatsby's reinvention is completely geared towards this objective.
The reader, therefore, comes to understand that the American dream is not just about wealth; it's about having the ability to create the perfect life. In this case, Gatsby's perfect life is all about winning back Daisy and picking up where they left off before the war.
In this novel, pursuing the American dream is a negative experience for everybody concerned. Gatsby, for example, never wins back Daisy and, in fact, loses his life when he is gunned down by Wilson. All of Gatsby's money and success could not help him achieve his dream of being reunited with Daisy or save him from Wilson's desire for revenge. Essentially, all of Gatsby's hard work comes to nothing.
Similarly, Nick returns to the Midwest after Gatsby's death because he has realized that the American Dream has a dark, destructive side to it. Remember that Nick had his own version of the American dream too, a dream in which he moves East and becomes successful as a result of his hard work.
However, after Gatsby's death, Nick refuses to pursue, or be a witness to, the American dream any longer. Having seen first-hand its destructive effects through the fate of Gatsby, he no longer wants to be caught up in this kind of life:
"After Gatsby's death the East was haunted for me like that, distorted beyond my eyes' power of correction."