From Exterminatus Manual
Each Exterminatus map has a series of objectives in it. Each team starts with control of it's own objective. Dead players are spawned at the objective controlled by their team and watch the view from their team's next objective while waiting for a spawn wave.
Some objectives can be captured by aiming at them and holding the USE key. Capturing an objective will reward you with points to spend on a better class next time you respawn. When an objective is captured the capturing team will begin to spawn there and a new next objective is set. Some objectives can be destroyed once the previous objective have been controlled. Breakable objectives give rewards for the damage done to them in the same way as wounding enemy players does.
The objective you should be attempting to control will be marked on your heads up display (HUD), this will always indicate the direction of the objective, even throgh solid walls. You can use the console command "ex_obj_show 1" to show the state of all the objectives or use "ex_show_obj 0" to hide them all if you find them distracting.
The Objectives on the current map and their statuses are listed on the Tactical Overview screen (which is bound to the F1 key by default). Objectives are shown there in the order in which they must be captured, along with the icon indicating the objectives owner. The Tactical Overview also restates your team's mission and presents information about your current player class.