Kosher Nostra

"What can I say about Lenny? Let me think... Oh yeah. He's a putz. And he's lazy. And he's greedy. But he's my wife's brother. How can you make a cake with those ingredients, huh? *Chuckles*... The Sheldon kid? Think I want him dead."

—Mickey Cohen: L.A. Noire

Organized crime needs some kind of criminal underclass in order to exist. Criminal underclasses need some kind of social underclass in order to exist. And what group, historically, has generally been a social underclass? Take a guess. Jewish mobsters are every bit as nasty as the other kinds, with the addition that they might have Yiddish as a first language.

In fiction, the Kosher Nostra are rare, probably because because Jewish organized crime tends to be rolled into other, larger organized criminal groups. Historically, that was the Italians. Nowadays, the Jewish mob is mostly focused in Israeli or Russian mobs. Still, they tend to pop up often enough in early-mid 20th century period pieces, particularly in Los Angeles.

The Jewish mobster in fiction is often characterized as a Motor Mouth, with a tendency to act polite and friendly until they get suddenly and horrifically violent. Comedic tropes applied to Jews usually are not applied to Kosher Nostra types - in other words, Jewish gangsters who argue and complain a lot (probably written by someone Jewish) are rare.

Compare The Mafia, The Mafiya, The Cartel, The Triads and the Tongs, Yakuza and The Irish Mob.

Examples of Kosher Nostra include:

Film

The character of Moe Greene in The Godfather is based on Bugsy Siegel.
The character of Hyman Roth in The Godfather Part II was based on Meyer Lansky
Bugsy, about... well, Bugsy Siegel. Lansky is in this one too; understandable, given they were lifelong friends.
Bugsy and Lansky also appear in Mobsters.
Marty Augustine (Mark Rydell) is one of these in The Long Goodbye.
In Snatch, the characters make several references to the involvement of Jews in the underground diamond trade, possibly a bit of Take That to De Beers.
Once Upon a Time in America - David, natch. Plus, the character of Maximilian "Max" Bercovicz was inspired by Lansky.
Bernie Bernbaum plays one of these in Miller's Crossing. Notable in that he's not part of a bigger mob, and likes to play Shylock to evoke sympathy before stabbing people in the back.
In the film Pi, some of these are after the all-powerful number found by the protagonist, with the belief that it is the True Name of God.
Lucky Number Slevin features a boss named The Rabbi.
Le Grand Pardon is basically a French and Jewish Black-Foot version of The Godfather.

Literature

The Yiddish Policemen's Union takes place in an alternate history where Israel never took off; there's a Jewish state within Alaska. An ultraorthodox sect called The Verboven are the equivalent of The Mafia there.
In A Conspiracy of Paper, one of the characters is a Abraham Mendes, a historical figure, who in reality, as in the novels, is the chief lieutenant of the gangster Jonathan Wild. The novels have a definite hardboiled detective feel (despite the 18th century setting), and likewise, Mendes fits a lot of the 20th century (Jewish) gangster stereotypes. He's a violent thug in professional life, but when not engaged in crime, is a pleasant man who goes to synagogue regularly. He also fits the erratic temper part. The second novel shows him to be an animal lover, and he brutally attacks someone who had harmed a dog.
The gangsters in Shea and Wilson's Illuminatus! trilogy are used to confound the expectation that all American gangsters are or were Italian-American. "Dutch Schultz" and Weissmann are clearly Jewish.
The Great Gatsby has Gatsby's shady business partner Meyer Wolfsheim, who is a pretty obvious No Celebrities Were Harmed version of Arnold Rothstein, right down to "rumored to have fixed the 1919 World Series".
E.L. Doctorow's Billy Bathgate is about Dutch Schultz, real life Jewish mobster. (Played by Dustin Hoffman in The Film of the Book).

Live-Action TV

Young Bugsy and Lansky are characters on Boardwalk Empire, which also has Arnold Rothstein, as well as a fictional character, Manny Horvitz, who could be accurately described as what Tevye would be like if he was an Ax Crazy psychopath.
A Jewish mafia existed in real-life early 20th century Odessa. A Russian TV Series, The Life and Adventures of Mishka Yaponchik, was shot about one prominent Russian-Jewish gangster from Odessa. The later-famous Red commander Kotovsky was also a member; there was a Russian TV series about him, too.
On The West Wing, Toby's father was a hitman for Murder, Inc..
The Sopranos had Hesh Rabkin, Jewish music producer who did business with the show's Italian mafia. He may have been based on music mogul Morris "Mo" Levy, a real-life Jewish producer with mafia ties.

Video Games

L.A. Noire has a couple of these, as you might guess from the page quote. The first is Mickey Cohen, based on the real-life gangster who was something of celebrity in L.A. during the 40s and 50s. The second is Lenny Finklestein, a drug dealer whom you wind up killing while investigating morphine-trafficking.
The 3rd Street Saints in Saints Row might be channeling the Purple Gang, a prohibition-era crew of alcohol runners. Like the old Gang, The Saints all wear purple, the default car is "The Bootlegger", and they're known for openly violent crimes and intra-gang violence.
Grand Theft Auto IV has the Jewish Mob responsible for the diamond deal that serves as the focus of a major subplot. The boss and his right-hand man (Isaac Roth and Mori Green respectively) are Shout-Outs to the aforementioned Godfather characters.