An Assassin or Hitman is a person (a vast majority of the time a gangster) that is payed to kill people. They play a considerable role over the course of the GTA games, and the player is usually an assassin themselves.
Claude, protagonist of Grand Theft Auto III, was formerly a bank robber (among other crimes) before his girlfriend Catalina turned on him and left him for dead. He then became a member of the Leone Family but was later chased out of town for having an affair with the Don's wife. He then became a member of the Yakuza, which included assassination. Then he started working for Ray Machowski, the Yakuza mole in the LCPD. Claude became an assassin for Machowski as well. He later became an assassin for Donald Love, friend of Machowski.
Tommy Vercetti was the personal assassin for Sonny Forelli, Don of the Forelli Family 15 years prior to the events in Grand Theft Auto: Vice City. Forelli eventually hired other assasssins to kill Vercetti himself (for unknown motives) but they were all killed by their target. Vercetti earned the nickname "The Harwood Butcher" and spent 15 years in prison. When he got out, he was transferred to Vice City, eventually started the Vercetti Gang, and ended up killing Forelli for his betrayal. Notably, he took the place of Leo Teal in Mr. Black's Assassination Ring.
After CJ's brother Sweet was sent to prison, he started working for Frank Tenpenny as an assassin. Tenpenny was a corrupt cop and CJ killed people that would expose him and his partner Eddie Pulaski. CJ went on to be an assassin for the Leone Family and Triads.
Being a member of the Leone Family, being an assassin is part of the job. Prior to the events in Grand Theft Auto: Liberty City Stories, Toni was forced into leaving Liberty City after killing a rival mobster for Salvatore Leone. Upon coming back, he resumed his work. Toni worked as an assassin also for Donald Love.
After being dishonorably discharged from the US Army, Vic started working for Phil Cassidy, Marty Jay Williams, Umberto Robina as an assassin and also starting his own gang. It seems he would always be assassinating Cholos and Bikers. Later in the game, he would chiefly assassinate rival Drug Lords under orders from the Mendez Brothers or just for his own power.
Huang's father was son of the Triad Leader, and therefore a Triad. Being any sort of gangster involves assassination.
Leo Teal was a chef in Ocean Beach, Vice City, but also a Drug Dealer and assassin for the mysterious "Mr. Black". Teal, as hinted by several phone calls, played a major part in the ambush of the Forelli-Vance Deal. After selling some of the drugs to Ricardo Diaz, he was killed by Tommy Vercetti when he wouldn't reveal any information.
Several unnamed assassins were employed over the course of the games. Ma Cipriani employed an army of assassins to kill her own son. They wore completely black suites and were bald with 5:00 shadow. Vincenzo Cilli employed Leone assassins to kill Toni as well, dressed as casual Leones (black dress suites with gloves, black goaties and slick hair). Normally, mafia assassins dress as their own gangsters.