(covers information from several alternate timelines)
A criminal organization, also known as a gang or the mob, was a group of people collectively engaged in criminal activities. Generally referred to as simply "the organization" by its members, these groups could be motivated by profit or political reasons.
In Earth's Ancient West, the Clanton gang were criminals who engaged in the famous Battle at the OK Corral in 1881 against the rival Earp gang. (TOS: "Spectre of the Gun")
In Chicago of the 1920s, when members of criminal organizations were sometimes referred to as gangsters, gangs almost took over from conventional government when the latter nearly broke down. One book that was published about the gang culture of that time was Chicago Mobs of the Twenties. (TOS: "A Piece of the Action") A similar gang-related publication was Chicago Gangs. (ENT: "Horizon")
In 1944 of an alternate timeline wherein the Nazis occupied much of the United States of America, a gangster called Sal told Jonathan Archer that Irish gangs were among numerous groups that were rebelling against the Nazis (other such collectives being dockworkers, black people and women). (ENT: "Storm Front")
Criminal organizations were a staple of fiction set on 20th century Earth. For example, a Jack-in-the-box feature in the Bashir 62 holoprogram involved Vic Fontaine's Las Vegas casino being take over by gangsters. (DS9: "Badda-Bing, Badda-Bang")
One form of gang is a rape gang, which is typical on the planet Turkana IV. (TNG: "The Naked Now", "Legacy")
In 2365, a gang comprised of Morn, Hain, Larell, Krit, and Nahsk conspired to pull off the Lissepian Mother's Day Heist. (DS9: "Who Mourns for Morn?")
The Iotian culture of Sigma Iotia II modeled their society on the organizations from that period found in the novel Chicago Mobs of the Twenties, left behind by the starship Horizon. (TOS: "A Piece of the Action")
The Orion Syndicate was an example of a 24th century criminal organization. They would undertake criminal activities designed to affect the politics of whoever was willing to make use of their services. During the Dominion War, they collaborated with the Dominion to assassinate a Klingon ambassador, which was set up to sow dissent within the Klingon Empire so Dominion forces could exploit it. (DS9: "Honor Among Thieves")
Criminal organizations typically did not tolerate betrayal, and would take action to prevent their members or anyone who made contact with them from going to the authorities. In 2373, Quark and Odo survived an assassination attempt from the Orion Syndicate, when Odo took Quark to Inferna Prime to testify before a Federation Grand Jury about the Syndicate's activities. (DS9: "The Ascent")