Watcher Mines are small explosive orbs that the Watchers use to protect installations that they do not have the manpower to protect. They are floating proximity mines, triggered whenever something flies too close to them. However, due to their very explosive nature, shooting the mines will take care of them fairly easily.
Later on in the game, there is a variant of the Mines used by Watcher Battleships as turrets; these turrets are concealed beneath the hull when not in use in order for their protection (as they are still mines, and therefore still highly explosive). These "turret mines" can also be launched in your general direction, and if they hit you they will explode, causing a massive amount of damage to Will; this makes them very dangerous if ignored.
These mines also apparently power the shields protecting the power cores of the Battleship, and when destroyed, the shields lower. Destroying these power cores (referred to specifically in-game as the weak point of the ship) will cause the Battleship to blow up in a ball of blue flame, indicating that you have beaten the mission.