Battlesight Zero is a military term that describes an aiming method where the soldier adjusts the weapon sights so that the the bullet rise and drop over its trajectory stays within the desired area of the target. The soldier "dials in" the sights of the weapon to hit the mark on a target at a relatively short (usually 25 or 50 yards) distance in a manner calibrated to correspond to a point at the beginning of the bullet's trajectory that mirrors the strike point at the end of the bullet's trajectory. In other words, by determining where the bullet strikes at the beginning if its arc one can determine where the bullet will probably land (not accounting for wind) at the end of the trajectory.
Once the soldier has "zeroed in" the weapon the sight adjustment can be memorized by counting how many clicks it took to get to the zero setting. This is an extremely important asset, as you can't take the rifle you used in basic with you to the battlefield. In a military weapon with iron sights the front pin adjusts top to bottom, the rear ring adjusts left to right. Once the soldier has established a Battlesight Zero for the way they hold a weapon and sight it, he or she can pick up any weapon of that type and by counting the clicks from bottom front and hard left or right rear to their memorized setting they can know with confidence that they have the weapon adjusted for their sight picture.
misterfuffie points out that, due to variances in tolerance from issues such as wear & tear, this can only provide an approximation. True, but 90%-plus accuracy is better than none. Battlesight Zero always ensures that a soldier can trust the weapon in his or her hands.