Nick Carraway is the narrator of The Great Gatsby by F. Scott Fitzgerald, and he is telling the story after the fact, upon reflection. This is the summer he spent with Jay Gatsby, a man who represented everything Nick hates.
On one level. the entire first chapter is about the American Dream. Nick's father has agreed to pay for Nick to have a year to study and learn the financial bond business. His father must have done well to be able to afford to do this, and Nick is working on the American Dream--working hard and living frugally so he will one day be financially successful, as well. We also have the Buchannans, a spectacularly rich family who clearly does nothing but enjoy their fortunes, though of course neither of them seems to be particularly content with what they have.
And then there is Jay Gatsby. While we do not learn a lot about him in this chapter, we do know several things which suggest he, too, has benefited from the American Dream. He lives in a stunning mansion in West Egg, the "less fashionable" of the two islets known as East and West Egg. This indicates that Gatsby is part of a group that is referred to as "new money," as opposed to those who have inherited their money in a chain of wealth. This is the result of pursuing the American Dream.
In another sense, there is the idea of personal hopes and dreams, and this is most clearly articulated by Nick when he describes Gatsby to us. He says Gatsby had something no one else he met that summer, or maybe ever, had:
it was an extraordinary gift for hope, a romantic readiness such as I have never found in any other person and which it is not likely I shall ever find again. No--Gatsby turned out all right at the end; it is what preyed on Gatsby, what foul dust floated in the wake of his dreams that temporarily closed out my interest in the abortive sorrows and short-winded elations of men.
The mention of hopes and dreams here is, unfortunately, also tinged with a foreshadowing of dying. Note the words preyed, foul, dust, abortive sorrows, short-winded elations. This suggests that the hope and dreams may not live for long in this novel, at least for Gatsby.