As kmcappello notes, Jay Gatsby is supremely concerned with living the American Dream. While working as a janitor, Gatsby is invited aboard a magnificent yacht; after just a brief glimpse into the life of the yacht's wealthy owner, Gatsby is determined to have such a life for himself.
More specifically, Gatsby becomes consumed by his determination to win the affection of Daisy, a woman he meets in Alabama. He is immediately intrigued by Daisy's popularity and wealth, and sets out to be with her. However, Gatsby's military obligations require him to spend time abroad, and when he returns, he learns that Daisy has married Tom Buchanan in Gatsby's absence.
Gatsby purchases a home in West Egg so that he can be near Daisy, and he throws frequent and lavish parties there in the hopes that she might attend. At one point, Nick notes that Gatsby pursues Daisy like "the following of a grail," and eventually, Gatsby and Daisy do reunite.
Though the pair do not end up together, Gatsby's purpose in life is to attain the American Dream so that he may be with Daisy. Impressively, he is not swayed from his path despite the fact that Daisy is with Tom. Thus, Jay Gatsby is one of the most persistent, driven characters in American literature, as his goal in life is to work hard to find happiness and live the American Dream.
In The Great Gatsby, Gatsby's sense of purpose centers completely around Daisy, and his past with Daisy. Anything else is important only as it contributes to his recapturing Daisy and their past together.
Gatsby is obsessed with what turns out to be an illusion. He spends his life trying to recapture a past that never was. He earns money so that he will be economically acceptable to Daisy, and will be able to compete with Tom for Daisy. He throws parties hoping that Daisy will happen in to one. He creates a persona, an aura of mystery about himself, an image, in hopes of capturing Daisy. That is his one and only purpose.
Unfortunately, Daisy hasn't been pining for him for the five years since they first met, and she did, at least at one point, love Tom. This ruins Gatsby's dream. For his illusion to be true, Daisy had to have loved him as much as he loves her. And that just isn't the case. Gatsby loves completely and obsessively and unfailingly, but Daisy does not. Gatsby's is a single-minded existence, but unfortunately his single-mindedness leads only tro tragedy.
In F. Scott Fitzgerald's novel, the narrator, Nick, tells the story of Jay Gatsby. As a character, Gatsby represents the American Dream. He began life poor, but became an incredibly rich man. We learn later in the book that Gatsby's fortune was not exactly legal; regardless, Gatsby worked incredibly hard to earn his fortune.
If Gatsby is a symbol for the American Dream, his sense of purpose is symbolized by the green light. Early on in the novel, Nick watches Gatsby gaze across the water at this light, which we later learn is the dock light of Daisy Buchanan's house (this is a great passage to quote from). It turns out that Gatsby fell for Daisy when she was a girl, but she rejected him in favor of Tom Buchanan, a young man from a very rich, very fine family. Gatsby never got over Daisy, and earned his money in order to win her back from Tom.