Conquest is a system that appears in Final Fantasy XI and it refers to the portion of the world each faction controls. There are four factions in the conquest system, one representing each of the three starting player cities (Bastok, San d'Oria, and Windurst) and one faction is for the beastmen, non-player races generally considered evil or unenlightened.
Players can speak with the gatekeepers of their cities to gain a Signet. Players who have a signet will gain points for themselves and for their nation when they kill a creature that grants them experience. For the players this means they can redeem, by spending these points, items from the same gate-keepers who grant the signet. For each of the three nations, the accumulated points will determine what nation will control the region the following week. The three nations are also ranked first, second, and third depending on how well they did the previous week.
This might not seem to have much effect on players, but there are vendors that sell particular things in each city depending on who controls a region. Some items can only be purchased, either from vendors or through conquest, when a city is in 1st or 2nd place. Many items also have specific effects that are active only when used inside a region that is in their home nation's control while others have a specific effect only in regions not under their home nation's control.
Players can also buy items from other nations with their own nation's conquest points so long as their nation is ahead of the nation they want to purchase the item from. The price, in conquest points, is doubled for non-residents.