The Triarch were the overall rulers of the Necrontyr Empire. As the name suggests, they were a council of three Phaerons. The most senior Phaeron was called the Silent King because he never spoke directly to his subjects; instead, the other two spoke for him.
While the Necrontyr Empire's Dynasties were largely independent, and basically ran their military and civil affairs as they saw fit, they were bound by the Triarch under ancient codes of behavior. To enforce these codes, the Triarch raised forces of Praetorians and Stalkers and sent them among the Dynasties to defend the Empire against the forces of rebellion and anarchy.
The pact with the C'tan placed all command authority with the last Silent King, Szarekh, leaving the other two members of the Triarch, Hapthatra and Mesophet, as little more than figureheads. They were later destroyed in the rebellion against the C'tan at the end of the War in Heaven; as of the 41st Millennium, the only remnants of the Triarch are Szarekh (who has only recently returned to the galaxy after stumbling over the Tyranids in his self-imposed exile) and his Praetorians, though it turns out Hapthatra and Mesophet survived the rebellion (probably got blasted and took a long time regenerating) and are still accompanying Szarekh.