Direct Combat is a fight which takes place when one unit attacks another unit right next to it. Most of the units in the game are direct combat units. Max is a CO that specializes in direct combat.
In direct combat, the attacker fires first, and then (if the defender is also capable of direct combat), the surviving defenders fire on the attacker. This applies in the Advance Wars games, but not the Japan-only games, where both units generally attack simultaneously--although Game Boy Wars 1/2/Turbo tries to alleviate this by having the attacking unit deal half an HP extra damage. Game Boy Wars 3 uses the Initiative system, where the unit with sufficiently higher Initiative gets to attack first.