Weight & Balance

Before we can fully work through the whys/how of a weight & balance problem, we need to define some

Terms:
Reference Datum - imaginary point or plane from which all horizontal distances are measured for balance purposes.  Usually located at nose or engine firewall.
Station - Location along fuselage given in distance
Arm - Distance from datum to CG of an item (related to station)
Moment - product of Weight x Arm.  Is a force expressed in pound-inches
Moment Index - Simply mathematical reduction of zeros to make math easier
Center of Gravity - Point at which airplane would balance.  Total moment / Total weight
Basic Empty Weight - Standard Empty weight + optional equipment that was installed
Payload - weight of occupants, cargo, and baggage
Useful Load - Difference between takeoff weight (or ramp weight) and basic empty weight

It is established that 1 Gallon of gasoline is 6 lb.
1 Gallon of Oil is 7.5 lb

Weight 
Weight is an obvious factor in aircraft performance.  A heavy aircraft will have a
-longer takeoff/landing roll
-higher takeoff speed
-reduced rate of climb
-shorter range
-reduced cruise speed
-reduced maneuverability
-higher stalling speed
-excessive weight on nose wheel

Where the weight is located is of great significance as well.  We are primarily concerned with the distribution of weight within the aircraft's CG range.

One wing tank being empty while the other is full is one example of lateral unbalance.

Loading the aircraft too far forward or aft creates a nose or tail heavy situation, which can prove difficult to control.  A tail heavy condition results in very light control forces and possibility of unrecoverable stall/spin.

14 CFR Part 91 Requires pilots to operate within the CG limits.

Weight changes must be updated if equipment is removed or installed.  Some items are considered negligible weight (less than 1 pound if aircraft empty weight is less than 5,000 lbs etc.)

Computations
Weight x Arm = Moment  and Total Moment/Total Weight = CG.   Ensure that these meet manufacturer's limits.

Graphs

In some cases, you may see a dashed box within the moment envelope, indicating "Utility Category."  This indicates that if loaded within that box, the aircraft may be approved for spins or other maneuvers.

Lastly, there is a Table Method available in some cases.

Remember as well that weight can shift during the flight.  This is why TWO weight-and-balance problems must be computed.  Fuel will be burned, and at 6 lbs per gallon, this is a significant change.

Remaining within the Weight & Balance Limits of an aircraft is critical to the overall safety of the flight.