The Trident is a race in which the Admiral of Limsa Lominsa is selected. While some specifics are left unknown, some basics are articulated by NPC text.
Though Trident most often refers to a three-pronged spear, the etymology simply means "three toothed," which in this case refers to the three sections of the race, which seem to be a circumnavigation of the lower half of Vylbrand. Anyone in Limsa Lominsa may participate, but, as the race is nautical in nature, it is generally the captains of ships who have the most credible chance of winning.
During the race, there are very few rules that must be observed, and competitors are even encouraged to attempt to sink each other's ships, even at the expense of the lives aboard. One must be willing to risk everything to become Admiral; as such, treachery abounds in many Tridents.