Nick Carraway identifies Jay Gatsby's defining attribute as 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." We see Gatsby's indomitable optimism throughout the novel. He starts out from a humble beginning as the son of a dirt farmer in the Dakotas, but he proves to be a young man of limitless ambition, energy, and self-esteem. As a boy, he follows a regimen of activities designed to improve himself. As a young man, he befriends a millionaire after saving the man's life and learns the ways of a gentleman. While while stationed in Louisville during the Great War, he meets a teenage Daisy Fay.
His pursuit of the young Daisy tells you much about what sets Gatsby apart. Any soldier of eighteen to twenty years of age might be naive enough to think he could win the hand of the most sought-after debutante in town. Gatsby, however, is so self-assured that between his uniform and his gentlemanly manners, Daisy has no inkling he is penniless and comes from a lower class background. Moreover, he doesn't fawn over her or harbor a secret crush like most boys who want a girl that is out of their league—no one is out of Gatsby's league.
There's more to Gatsby than just confidence and good table manners, though. He is a man so transported by his own ideals and vision that he comes across as unusually earnest—a rare quality in decadent times. Jordan Baker, witnessing Gatsby's courtship of Daisy, describes this effect: "He looked at her the way all women want to be looked at by a man." Nick Carraway is also quite impressed when he first meets Gatsby:
He smiled understandingly-much more than understandingly. It was one of those rare smiles with a quality of eternal reassurance in it, that you may come across four or five times in life. It faced—or seemed to face—the whole eternal world for an instant, and then concentrated on you with an irresistible prejudice in your favor.
Needless to say, the young Gatsby sweeps Daisy off her feet. However, he must leave for the war. Such is the impression that Gatsby has made on her that Daisy is actually willing to wait for him, and she only marries Tom under pressure from her family.
Meyer Wolfsheim also sees Gatsby's potential immediately—but in the context of business. He is an ideal front for bootlegging: he is smart, hardworking, trustworthy—a good business partner—but he also has the bearing of a true gentlemen, while Wolfsheim and other gangsters do not. During the scene in the speakeasy, we see that Gatsby is friendly with the most powerful men in New York, and it's clear he is so well liked that no one is going to crack down on his empire anytime soon.
So we see that a combination of attributes accounts for Gatsby's rise and his success wooing Daisy. He is a man of great vision who never doubts his ability to realize that vision. He is smart, energetic, charming, and extremely earnest. Gatsby also possesses a single-minded focus on his goals: he is capable of pursuing his aims for years without tiring or losing hope.
The principle flaw of Gatsby's personality is his willingness to overlook and even edit out information that doesn't fit with his ambitions. He changes his name and invents a backstory for himself that is so rehearsed it convinces no one. After Gatsby fails to win Daisy Buchanan away from her husband, Nick tries to gently impress upon his neighbor that he can't recreate the romance they once had. Gatsby is undaunted: "Can’t repeat the past?…Why of course you can!" He considers his failure to be just another setback. He has overcome so much, after all; what's another obstacle? He fails to recognize, however, that Daisy lacks the courage to leave her husband as well as the stamina to live up to Gatsby's vision of her. What's more, he's so focused on his next move to win Daisy that he completely fails to register the danger that Myrtle Wilson's death represents.