What was the role of Meyer Wolfsheim? Why was it so important?

I am not acquainted with the book myself. But as far as the movie is concerned, it is not expected to be known much about Wolfsheim.

Wolfsheim is a friend of the Gatsby. He is a member of New York's underworld. He also fixed the 1919 World Series. He was in some sort of "bond-scam". Further, Wolfsheim gave some hints to Nick that seem to indicate that Gatsby, who always tried to beat his own drum in front of the public by throwing parties and all, was merely an associate of the former and Wolfsheim carried much of the later's expenditure in order to carry on with his own business making Gatsby a mere carrier of his interest. As The Washington Post describes-

And we have to assume, I think, that Gatsby's home and his parties are, to a greater or lesser degree, a business expense for him. They're part of creating the Gatsby vehicle that can carry the Wolfsheim business.

Wolfsheim seems to be using Gatsby for retaining his own anonymity and he might as well was funding fir the parties to get familiarized with the high-society. Although, the sponsor can the Gatsby too; this remained a mystery.

So as a character, Wolfsheim was not that important, but he let us know some true facts about the Gatsby, who apparently could not be believed easily regarding his business profile.

To add to the other answer,

It sounds like Wolfsheim may have been a reference or a parody to the real life Arnold Rothstein, who was a "fixer" and the Kingpin of New York City Jewish Mob during the prohibition era and Jazz Age. He did allegedly fix the 1919 world series!

(A fictional version of the character is featured on the HBO series Boardwalk Empire played by the wonderful Micheal Stuhlbarg)

The reason though the idea of character is significant is because it tells us that these are kinds of people (ie: the mob) that Gatsby had become associated with in order to secure his cushy life style.

It effects the story in a couple of ways:

1. It's about the theme. Fitzgerald is telling us that wealth, especially acquiring it by these means, has consequences, but more over that it doesn't grantee one's happiness since Gatsby dies and Daisy leaves with her somewhat monstrous husband, granted she has a child.
2. That even though the consequences of what happens to Gatsby are not directly related the mob, the idea of wealth inequality, political corruption, or "mob mentality" are still on display, as Daisy's husband is a monster, Nick thoughtlessly spills the beans to Myrtle's husband about Gatsby's car, and Gatsby is willing to take the fall for his rich, beautiful, but seemingly careless or ignorantly reckless Daisy, thinking it was ok to do nothing about Daisy running over Myrtle, saying , 'he would take care of it'.
3. That by possibly connecting his story to the New York City's [Jewish] Mob, Fitzgerlad gives us some real insight and cultural tether to New York City's darker version of the Jazz Age.