You control a 
province, if at the same time:
Influence in a province is the total number of 
hit points of all your cities ( 3), castles ( 4) and 
military units ( 1 or 2) present there.
Mike controls province A, because he has a city there and his influence there equals 7 (3 for city + 2 for swordsmen + 2 for galley). This number is greater than the 2 influence points that Alex has (1 for archer + 1 for swordsman).
Mike does not control province B, even though he has a castle there. His influence in that province is 5 (4 for castle + 1 for archer), which is less than Alex's influence of 6 (3 for archers + 1 for catapult + 1 for swordsman + 1 for cavalryman).
Although only Alex has influence in province C, he does not control it, as he has not built a city or a castle there yet. Therefore, no one controls the provinces B and C.
During the achievement phase, each province you control grants you Glory and coins.
Many military achievements involve area control, which is one of the pathways to victory.