Weight-and-balance is determining that an aircraft is loaded within the limits of it's weight carrying capability and that the center of gravity is within the operational envelopes of the aircraft.
Weight-and-balance typically refers to insuring that the aircraft is loaded so as to be within the weight limitations and the balance limitations for intended operations. Commercial operators are usually required to complete a weight-and-balance form before takeoff. That form typically must then be retained for a period of time and available for inspection by regulating authorities. Weight is not just the overall weight limited by aircraft performance, but also limits imposed by the aircraft structure. For large aircraft, there is a max weight for taxi, another slightly lower for takeoff, another for max zero fuel weight, and another for max landing weight. Each of these maximums may also include separate operating envelopes for each condition that express the forward and aft limits of the center of gravity.
The center of gravity is typically expressed in percent of the mean aerodynamic chord, often shown as %MAC. Large aircraft typically have a recommended %MAC for the zero fuel weight for optimal operaton.