The Volturi coven was founded by Aro. When the coven was strong enough, the Volturi usurped control of the vampire world from the Romanian Coven. At some point the Romanian coven tried to regain control, but the Volturi defeated them.
The Volturi have ruled the vampire world for centuries by the start of the series. During that time, they did a number of things to ensure stability in the world, including eradicating the Immortal children and cleaning up after the Vampire Wars in the southern parts of America. At some point, Caius was almost killed in a fight with a werewolf. Becoming terrified of his people's mortal enemies, he pushed the Volturi to hunt them to near extinction in Europe and Asia.
At one time, Marcus and his wife Didyme, who was also Aro's sister, planned on leaving the Volturi. Aro was upset by this, but appeared to give them his support. However, Aro later secretly killed Didyme to prevent Marcus from leaving.