A battle is a prolonged combat involving large armies. An engagement may be a general encounter between armies, or a minor encounter between smaller forces. A skirmish is a small encounter usually incidental to a larger military movement, while an action stresses the idea of active, frequently sharp, offensive and defensive movements. The term affair simply means a fight.
I've seen another set of divisions that defines the combat type based on troop size, i.e. a battle involved an army from each side, an engagement a corps from each sided, and on down the line.