It's important to note that Nick Carraway, our omniscient (and somewhat unreliable) narrator throughout this novel, is never explicit about what Gatsby did. It's because he didn't really know either. We're led to think, through certain context clues, that Gatsby made his fortune in bootlegging and Wall Street speculation. Both pursuits were common ways of acquiring wealth in the 1920s.
The bootlegging is brought up by Tom Buchanan, Gatsby's husband and rival for Daisy's affections. Buchanan also works well in the story as an example of "old money," in contrast to Gatsby -- an upstart from a simple, modest background.
Buchanan reveals that Gatsby did business with a guy named Meyer Wolfsheim. He says that they ran "'drug-stores' here and in Chicago and sold grain alcohol over the counter." The name "Meyer Wolfsheim" has a number of connotations: "wolf" suggests someone predatory; and "Meyer" is a reference to the associate's Jewish heritage. Bootlegging was considered a dishonorable way to make a fortune (though we know now that many prominent people became wealthy doing it). Also, upper-class white Americans did not generally associate with Jews in the 1920s. All of this is revealed to highlight Gatsby's status as an outsider, despite his immense wealth. Buchanan, during his exposure of Gatsby, refers to him as a "swindler."
In the novel, much of Gatsby's life and career remain unclear. Nick Carraway does not begin to learn much of the truth about him, including Gatsby's claims of being an Oxford alum, until the end of the novel when Gatsby's father, Mr. Gatz, comes to claim his son.