Stefan and Vladimir once held the position the Volturi do now--a ruling class--until the Volturi burned down their castle and became the ruling power fifteen hundred years ago. The Romanians have since held a vendetta against them (Breaking Dawn, pg. 658).
In Breaking Dawn, they arrive in Forks, uninvited, in the hopes of finally getting their throne back, but leave dissatisfied with the result of the confrontation.
Classic Quotes--
"We sat still for a long time, child. Contemplating our own divinity. It was a sign of our power that everything came to us. Prey, diplomats, those seeking our favor. We sat on our thrones and thought ourselves gods. We didn't notice for a long time that we were changing--almost petrifying" (Breaking Dawn, pg. 631).
"Now the Volturi's eyes are filmed with dusty scum, but ours are bright. I imagine that will give us an advantage when we gouge theirs from their sockets" (Breaking Dawn, pg. 631).