Conquest is an ability that can be used every 30 seconds by Towers, but the time restrictions can be decreased by chopping down Forests. It can be used to claim 4 tiles in a straight line beside the Tower on any of its 6 sides, so long as they are not blocked by an enemy or neutral Mountain Tile, Water Tile, or another Tower. It can capture Camps and activate them, capture capitals, and capture all other tiles.
In order to activate this ability, you must select one of your own towers where it will charge up. If this is interrupted by another player by attacking the tower, then the charge up will be cancelled and the ability cooldown will start over. Furthermore if you interact with another tile while the conquest charging is in progress, then the same result will occur.
Once the ability is ready, you can click on the tower and select any of the 6 directions of the hexagon where you want the conquest to occur. The selected regions will slowly fade into your control, the conquest ability once obtained cannot be stopped from activating unless the tower itself is captured.
You can transfer conquests between towers. By putting towers in the line of sight of each other, you can activate the ability of one tower and direct it at another tower to make the other tower have the conquest ability without having to charge up.