We will want to look at the characters of Nick and Gatsby to answer this question about The Great Gatsby.
First, Nick relates this narrative as a piece of his peronal history. He is telling us this story long after the action has taken place, looking back and recalling "how things were". This is a story, in part, of how Nick came to become the person he is. It is a story of formation, of personal development for Nick, as well as Gatsby.
Nick tells us that he had intentions of reading great books and of becoming a man of culture in the summer he rented the cottage next to Gatsby's mansion. This was Nick's naive dream. And we learn of this dream well before we learn about Gatsby and his dream.
So, in the first chapter, Nick presents us with a character with a dream (which is clearly not going to come true) and a person who is interested in delving into the past, to find some truth in it.
Soon, we meet Gatsby, a character who is defined by two things - a lie and a dream. He dreams to marry Daisy and achieve great wealth (of which Daisy is one part). He dreams, in short, to be a great man.
He is also a criminal who has changed his name and changed his history in order to become "Jay Gatsby". This is his lie.
His dream is hatched before his lie, when he has the vision of the white ladder reaching up to the heavens. And it is the dream that leads, almost directly, to the lie. Gatsby sets out to be a new and better person, to earn Daisy, by leaving the name Gatz behind.
For Gatsby and for Nick, the dream is unattainable because the dream too is a lie, to some extent. Gatsby never truly changes, as evidenced by the crowd at his funeral. Attended by only his father, essentially, Gatsby's funeral shows that his past was not left behind but was instead waiting in abatement. It comes back to him, or he goes back to it, in the end, and Nick does the same when he returns to the Midwest at the story's end.
The dream of transformation (into wealth, into "culture") is attached to a future that never quite arrives. And the past is always waiting to reclaim the characters in The Great Gatsby, as Nick suggests in the novel's final lines:
So we beat on, boats against the current, borne back ceaselessly into the past.