While Nick Carraway (carried-away) offers a romanticized portrait of Jay Gatsby, Wolfsheim reveals the sordid side of Gatsby's life. Wolfsheim's relationship with Gatsby--they are involved in illegal business schemes together, as Tom will later reveal ("He and this Wolfsheim. . . sold grain alcohol. . . I picked him [Gatsby] for bootlegger the first time I saw him and I wasn't far wrong") -- offers the reader a glimpse at Gatsby's criminal background and suggests that Gatsby, like Wolfsheim, has a hard streak. By linking the two, the novel also reinforces the idea, aired by Tom Buchanan early on, that the world is divided into the "Nordics," people like Jordan, Nick, and himself, and the "coloreds," a category that apparently includes all non-Northern Europeans. We learn from Nick that Wolfsheim is a "Jew." Whether or not Gatsby, born Gatz, is Jewish, is unclear, but Tom will later jeer at the idea of Gatsby and Daisy marrying as akin to interracial marriage.
The first meeting between Wolfsheim, Gatsby, and Nick also foreshadows Gatsby's death, as Wolfsheim immediately launches into a story about his associate Rosy Rosenthal being shot to death in front of the Metrople, mistakenly thinking Nick is a connection Gatsby wants him to meet. Clearly, Wolfsheim is part of the underworld: Gatsby calls him a gambler and confides to Nick that he fixed the 1919 World Series. This provides more than a few hints that Gatsby's rapidly acquired fortune rest on illegal gains.
The portrait of Wolfsheim also suggests that without his charm and yearning, idealistic dream of regaining his first love and all she represents, we might see Gatsby as just another common criminal, a sleazy con artist. This adds nuance and complexity to Gatsby's character. Even more interestingly, however, it is not the underworld but involvement in the ruthless upperclass world of the Buchanans that leads to Gatsby's murder. Fitzgerald seems to suggest that the world of inherited "Nordic" wealth may be even more dangerous, at least to people like Gatsby, than Wolfsheim's criminal underworld.
The rhetoric of Wolfsheim's name is significant: "wolf" as a dangerous carnivore and predator, and then "sheim" signifying German, which in the mid 1920s when Fitzgerald wrote Gatsbywas still synonymous with "enemy." It is interesting that Fitzgerald integrates that name linked with German character / enemy as part of (the underbelly of) the American dream.
The connection with this underworld character reveals a lot of missing pieces about the how, what, and why of Gatsby. Wolfsheim is the underbottom of success, the seedy connection that has made Gatsby his fortune at the price of his soul.
The character analysis of Wolsheim here at eNotes reads, in part:
"One of Wolfsheim's notable characteristics is his wearing of cufflinks made of human molars. He is so selfish and insecure that he refuses to attend Gatsby's funeral. Nick sees the gangster part of Gatsby's life as one of the ways he made his money, but he separates Gatsby's character from true insensitive, subhuman criminals like Wolfsheim. Gatsby stands by Daisy when she commits a crime, but Wolfsheim will not honor his relationships."