Jay Gatsby, the title character of The Great Gatsby, is not "great" in a moral or virtuous sense. He is an unrepentant criminal, bootlegging during Prohibition to create his wealth and pursue his dream, Daisy.
While he is seen as great by the community, who are wooed by his parties and his mysterious past, the inner person of Gatsby is in fact troubled. He creates a persona to fool people, declaring himself a war hero and changing his name. In the sense that he is a performer, he could be ironically called "The Great Gatsby" as a magician on stage; he creates a grand illusion to fool people. Nick, the narrator, sees him as an embodiment of the American Dream; he came from poverty into wealth and now lives a high-class lifestyle. However, Nick is blinded by his own idealism and does not consider Gatsby's criminal vocation or his adulterous affair to be factors. Gatsby cannot be seen as truly "great," and in fact the title is somewhat sarcastic, as his name is not really Gatsby either.
Interestingly, F. Scott Fitzgerald did not like the final title. He had several literary titles in mind, but he was overruled by his editor and his wife, who both thought his chosen title was too obscure.
In one respect, Jay Gatsby truly is great. In Chapter Six, for example, it is revealed that Jay Gatsby is really James Gatz, the son of some "unsuccessful farm people" from North Dakota. After working as a clam-digger on Lake Superior, Gatz changed his name to Jay Gatsby and, in the course of a few years, transformed himself into a wealthy and successful entrepreneur.
But this level of greatness does not extend to other parts of his life. His inability to win back Daisy Buchanan, for example, is a sign that Gatsby is not as great as the title suggests. In fact, her refusal to leave her husband implies that Gatsby's greatness is merely an illusion. Moreover, the fact that his empire is built on illegal and immoral dealings with the criminal underworld shows that his greatness has a much darker side.