Because of the streamlined shape of the airplane, loads carried inside tend to be close to its centerline. Placing a passenger in the left seat or the right seat is not going to affect how the airplane flies with any significance.
Fuel is carried in the wings, further away from the airplane's center of gravity. If there is a fuel imbalance between the left and right wing fuel tanks, the center of gravity can move significantly to the left or right. Pilot's must make sure the fuel load is balanced between the two wings, and remains balanced during the flight. This is accomplished through following normal procedures established for the airplane, such as remembering to switch fuel tanks every so often.
Fore and Aft Balance
The lift generated at the wing can be considered to be acting entirely at one place on the wing. That place is called the center of lift.
If the center of gravity is forward of the center of lift, a nose down tendency results during flight. The further forward the center of gravity, the stronger this nose down tendency becomes. Tail down force is continuously required from the horizontal stabilizer and elevators to counter this nose down tendency.
Should the center of gravity lie aft of the center of lift, a nose up tendency will exist. This nose up tendency becomes stronger as the center of gravity moves further aft. Tail up force is required from the horizontal stabilizer to counteract this tendency.
The location of the CG with respect to the center of lift determines the airplane's handling characteristics along the pitch axis, or its longitudinal stability. If the CG is too forward or aft of the center of lift, the airplane becomes difficult to control and unsafe to fly.
The airplane is normally loaded to place the center of gravity very near the center of lift. The airplane manufacturer provides center of gravity limitations to guide the pilot in ensuring normal performance and controllability.
Most light, general aviation airplanes are designed with an asymmetrical airfoil, a wing which is more curved on the top and less curved on the bottom. This is a more efficient wing design for aircraft that are not intended for maneuvers like inverted flight. With an asymmetrical airfoil, the center of lift (or center of pressure) will move forward as the angle of attack increases. It moves rearward with decreasing angle of attack. This backward and forward movement of the point at which lift acts affects the aerodynamic balance and controllability of the aircraft somewhat.
We pilots control the balance and controllability of the aircraft by ensuring the force of weight, acting through the center of gravity, remains in a good location for the entire flight.
The Nose Heavy Airplane
If the airplane is nose heavy, it tends to fly with more dart-like characteristics. A dart has an extremely forward CG. As a result, it is very stable and flies in a straight line well. If the dart's tail swings out of line, it will have only a minor effect on the its flight path.
The trade off with an airplane is maneuverability. The increasing stability that comes with a more forward CG makes it more difficult to alter the airplane's flight path.
The Tail Heavy Airplane
The heavy end of an airplane likes to go in front. The further aft the CG is, the more the airplane will take on an instability in the way it flies. While more maneuverable, the more unstable, jerky, and lighter feeling flight controls are undesirable characteristics in a general aviation airplane. The airplane is more difficult to control, especially in turbulence, meaning the increased chance of the pilot accidentally overstressing the airplane. A tail heavy airplane also has the very undesirable characteristics of being more difficult to recover from a stall or spin.
Fighter jets are designed with aft CGs, in order to make them highly maneuverable. The instability in these aircraft is an acceptable trade off of the design, and sometimes computers are even required to aid the pilot in controlling the airplane.