Direct Combat is combat which takes place when one unit attacks another unit next to it. Most units in the game are direct combat units.
Max is a CO that specializes in direct combat, while Grit is one that's weak in it.
In Direct Combat, the attacker fires first, then the defending side, if they haven't been wiped out, will counterattack. Indirect units can't counterattack, however. This applies in the Advance Wars series, 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.