The Area of Operations (AO) is the general area in which players are currently present and/or fighting, and may include the area of the objective(s) and/or enemy forces. As units move, the "AO" may shift along with them.
The term is typically used by Aircraft to indicate whether they are in the vicinity of the ground forces, leaving that vicinity, or approaching it. For example, an aircraft returning from base to the battle may indicate that it will arrive in the "AO" in two minutes. The term has little use outside that context.
Units inside the "AO" are active and under potential threat. Units outside the AO are relatively safe (barring a completely unexpected attack).
The Area of Operations does not directly correspond to the non-shaded area of the in-game map. That area only marks as the boundaries that players are not supposed to cross (and are unlikely to cross) during the mission, and often contains the home base which is not normally considered to be in the "AO".