One of the main themes of The Great Gatsby is the idea of the American Dream - what it is, how to achieve it, what happens when one achieves it (or doesn't), and what it implies for America. Through the characters, we see starkly different versions of the American Dream. Here are some quotes to highlight those ideas:
Nick Carraway, for example, seems to have a more traditional view of the American Dream in that if someone works hard enough, they can pull themselves up from their bootstraps and be successful. Based on the description of his family, this is certainly what the Carraways have accomplished (chapter 1):
The Carraways are something of a clan, and we have a tradition that we're descended from the Dukes of Buccleuch, but the actual founder of my line was my grandfather's brother, who came here in fifty-one, sent a substitute to the Civil War, and started the wholesale hardware business that my father carries on to-day.
In chapter 1, Nick also gets his very first glimpse of Gatsby, who is reaching his hand across the bay. As the novel develops, we come to recognize that the green light that he was reaching for in chapter 1 is the symbol of his American Dream - that is, recapturing Daisy's heart so that everything can be just like it was:
But I didn't call to him, for he gave a sudden intimation that he was content to be alone—he stretched out his arms toward the dark water in a curious way, and, far as I was from him, I could have sworn he was trembling. Involuntarily I glanced seaward—and distinguished nothing except a single green light, minute and far away, that might have been the end of a dock. When I looked once more for Gatsby he had vanished, and I was alone again in the unquiet darkness.
As Nick gets to know Gatsby, he describes him in astonishing detail. The way in which he describes Gatsby contrasts how he describes himself and his upbringing as well as those who are from old money. It's as if Nick unintentionally categorizes the traditional idea of the American Dream as more worthy or honest than the new money form of the American Dream (chapter 4):
He was balancing himself on the dashboard of his car with that resourcefulness of movement that is so peculiarly American—that comes, I suppose, with the absence of lifting work or rigid sitting in youth and, even more, with the formless grace of our nervous, sporadic games. This quality was continually breaking through his punctilious manner in the shape of restlessness.
About halfway through the novel, we could argue that the idea of Gatsby's American Dream vanishes forever once Daisy and him are "reunited," so to say. This makes us wonder what Fitzgerald is trying to say about the American Dream in general. Gatsby describes how he can almost see her house across the bay, and Fitzgerald writes (chapter 5):
Possibly it had occurred to him that the colossal significance of that light had now vanished forever. Compared to the great distance that had separated him from Daisy it had seemed very near to her, almost touching her. It had seemed as close as a star to the moon. Now it was again a green light on a dock. His count of enchanted objects had diminished by one.