Fitzgerald connects the American Dream to Gatsby's quest to regain Daisy. In both cases, Fitzgerald says, the dream is to start over again and remake the past. As Nick states near the end of the novel, he can understand how the new continent must have looked to the first settlers: "a fresh, green breast of the new world." They were tantalized by the dream of leaving the mistakes of Europe behind and beginning again, this time getting it right, setting history straight.
Gatsby looks like he has achieved the American dream of wealth and success, but that is not what he cares about (and he is scorned anyway by old money people like Tom): Gatsby's dream is to recapture the past through Daisy.
Gatsby, separated from Daisy for five years, wants nothing more than to go back and pick up where they left off, as if the intervening five years never happened. Of course, as Nick tries to tell him, that's impossible—too much has happened. Gatsby, however, refuses to hear that. He wants to set history straight and get it right this time with Daisy.
Likewise, the American Dream of going back to the beginning and building a paradise in "a fresh, green" new world can't happen. Too much has happened, and we carry too much baggage, yet we are tantalized by the dream of starting over. We race towards the future, thinking that one day we can achieve the dream of paradise. Instead:
the orgastic future . . . year by year recedes before us. It eluded us then, but that’s no matter—tomorrow we will run faster, stretch out our arms farther.