"Battle School" is set up as a combined military training and psychological assessment facility. Both Peter and Valentine were initially considered but ultimately rejected due to their unique personality traits; Peter for his borderline psychopathy and Valentine for her excessive empathy, neither of which are especially desirable in a future military leader.
The facility ultimately serves three purposes, each with an element of psychological profiling;
Honing the skills needed by the commander of the fleet, namely Ender Wiggins.
Ender (and his future commanders) are taught maths, science, self-defence and, through the "Battle Game" the rudiments of three dimensional space combat. Ender and his cohorts must feel confident in their abilities.
Providing the right psychological setting to allow Ender to mature as an individual.
The system computer (through the "Giant's Game") regularly pushes him to consider his personal makeup. On at least one occasion it forces him to use violence to overcome an apparently insurmountable obstacle and on another it makes him contemplate the difference between himself and Peter. The directors of the facility are looking for different things in different students. Some are being groomed as potential lieutenants, others are kept on as cannon fodder, still others (such as Bonzo) are there to provide Ender with a foil for his personal development.
Allowing Ender to gain experience of command
Ender must learn to trust his junior officers. Colonel Graff and Major Anderson regularly debate the decision to pervert the Battle School (by scheduling unfair scenarios) purely to allow Ender to become a human weapon but their personal feelings are trumped by the fact that the Human Fleet is approaching the first of the Bugger worlds.