The Triggermen are gangsters who base themselves around Boston's pre-war mafia families and their traditions. This appears to be due to the influence of their ghoul members, who claim to have been members of such organizations. Because of this, most Triggermen appear to operate with a code of honor, for example, Skinny Malone refuses to kill Nick Valentine because of the "old times", when they fought with each other in the past. They can often be seen in the company of Raiders and other such unscrupulous characters, running enterprises such as the Robot Racetrack at Easy City downs.
Triggermen live outside society, establishing secret bases where they undertake their questionable activities. One example is Marowski's chem lab, which is hidden deep in an abandoned fishpacking plant, and is where Marowski's gang create chems to smuggle into Diamond City and other settlements. Goodneighbor in particular appears to be a haven of many Triggermen gangs.
At the conclusion of The Big Dig, there are two Triggermen with Hancock's bodyguard Fahrenheit. This is probably why when White Chapel Charlie asks the player to clear the warehouses in Goodneighbor which contain Triggermen, he doesn't want to tell you that Hancock is the one behind his request. Since Hancock is having members of a group killed, he apparently works with from time to time.
Triggermen only appear in Fallout 4.