The Centre of Gravity of a body is the theoretical point at which the entire weight of that body is assumed to be concentrated.
In an aeroplane, the centre of gravity (CG) is the point at which the aircraft would balance were it possible to suspend it at that point. As the location of the centre of gravity affects the stability of the aircraft, it must fall within specified limits that are established by the aircraft manufacturer. Both lateral and longitudinal balance are important, but the primary concern is longitudinal balance; that is, the location of the CG along the longitudinal or lengthwise axis.
The empty weight and the empty weight centre of gravity (EWCG) are calculated for each aircraft using a set of weigh scales, the manufacturer defined reference datum, the arms, as listed on the aircraft Type Certificate Data Sheet, for each weighing point. The empty weight of the aircraft is simply the sum of the weights from each of the weighing points. The EWCG is calculated using the calculated moments for each of the weighing points, the calculated empty weight and the appropriate formula for the aircraft type as specified in the Aircraft Flight Manual (AFM). On small airplanes and on helicopters, the center of gravity location is identified as being a specific number of inches from the datum and the center of gravity range is identified the same way. On larger airplanes, the center of gravity and its range are typically identified in relation to the width of the wing. Thus empty weight centre of gravity may be expressed as a percentage of Mean Aerodynamic Chord (MAC) and is often referred to as the "basic index". If equipment is added to or removed from the aircraft after the weighing process, a revised empty weight and EWCG can be calculated mathematically by calculating the moment associated with the change and then modifying the basic index.
For day to day operations, the basic index is used as a starting point for CG calculations. Weight and arm information for the total aircraft load (crew, passengers, freight, fuel, catering etc) are used to calculate moments. The total moment for the load is used to adjust the basic index resulting in the takeoff centre of gravity. Centre of gravity calculations may be done mathematically or graphically at the discretion of the operator.
ICAO Annex 8: Airworthiness of Aircraft.
US FAA Aircraft Weight and Balance