The battles will be solved one at a time. First the attacker hits one unit at a time. Then the defender hits one unit at a time. The military units will then decrease with the hits taken on both sides. This will go on until all the attacking or defending units has been destroyed. You should see <l:h_hitpoints> for how strongly units hits other units.
The attacker only wins the area if he has conquest units (see <l:h_conquest_unit_land> and <l:h_conquest_unit_sea>) left in the area after the defending units have been destroyed. If for example only <l:eu_airplane> are left they cannot hold the area and will return to their airbase.