The most famous mafia, of course, was the Italian mafia. There's also the Russian mafia, Serbian, many others. Did the Jewish mafia conflict with these other organizations? What was the organization of the Jewish mafia? Was it a single organization, or a collection of separate criminals? Are they still around, or was it mostly near the beginning of the 20th century?
The term "The Italian Mob" can refer to a lot of different organizations (Sicilian LCN, Camorra, Ndrangheta, many more).
I'm assuming you're referring to American La Cosa Nostra, which the New York Five families, Chicago Outfit and most Traditional Italian-American organized crime are part of.
American LCN was based on Italian criminal traditions, but it was never really a purely Italian enterprise. While the Five Families in NYC technically only let 100% Italians make their bones, there were lots of non-Italian associates who were for all intents and purposes full freight members. Jews were probably the most numerous and definitely the most influential non-Italians in the Five Families (Irish criminals were also very well represented).
Meyer Lansky was a Jewish gangster who was born in Poland and raised on the lower eastside of Manhattan. Along with childhood friend/gang-homeboy Charles "Lucky" Luciano he basically created what we think of as "The American Mafia" by uniting a diverse, feuding array of ethnic bootlegging gangs into a well organized, hierarchical structure that was enforced by hardcore wiseguys.
Lansky, among other things, was a proponent of keeping the old school pseudo-catholic, cult-like initiation rituals of the Mafia alive. Luciano thought they were stupid, but Lansky knew that they would help build the allure of the organization. Lansky was a stone killer and great organizer, and is probably the main guy you should research if you are interested in Jewish wiseguys.
Lansky's (much crazier, less successful) friend/protege Benjamin Bugsy Siegel is also an interesting Jewish mobster. He was one of the leaders of a loose group of Italian and Jewish hitmen known to the press as Murder Inc.
Both these guys (and a whole lot of other Jews, and some Irish and other random guys) were never technically made members of the Five Families, but unofficially were in their highest echelons of leadership. I believe the non-New York LCN families had similar situations.
As for the eastern European mobs these groups are VERY diverse (Among the media and law enforcement The "Russian Mafiya" doesn't refer to 1 organization, but rather to any group of Russian speakers involved in nearly any crime). Russian Organized Crime groups currently have Jewish members, and Russian-speaking OC is very influential in Israel.