How do I determine the center of gravity on aircraft?
You ask particularly about the Center of Gravity (CG), but not the weight. Any CG calculation will necessarily include the weight as a component of the calculation; therefore I refer to the calculations as weight and balance (or CG) calculations.

The two primary ways that weight and balance measurements are determined are by physically weighing the aircraft and by mathematically calculating based on additions or subtractions to the previous weight. The former is performed by maintenance personnel and forms the empty weight and CG for any given aircraft. The later may be performed by either flight operators or by maintenance and uses the predetermined empty weight and CG as the basis for the calculations; this may be needed due to the installation of new equipment, for example, or a pilot may do such a calculation to determine the weight and CG for a given flight with, for example, fuel and passengers.

For maintenance operations, most aircraft have a section in the maintenance manual or operating handbook that prescribes the process. For flight operations, most aircraft have an information manual or operating handbook that will prescribe the calculation.

One necessary parameter is an established reference datum.

The basic formula is:

$$
C_G=\frac{(W_1\cdot A_1)+(W_2\cdot A_2)}{W_1+W_2}
$$

Where:


$W_n$ is the weight of the aircraft scale's measurement, predetermined empty weight, fuel, or payload, etc.
$A_n$ is the moment arm, in reference to the datum, of the of the aircraft scale's measurement, predetermined empty CG, fuel, or payload, etc.
$C_G$ is the solution to the center of gravity for the aircraft.


The cancelled FAA Advisory Circular AC65-9A contains an entire chapter (Chapter 3) dedicated to the subject of weight and balance, including exactly what you ask, the parameters that must be considered and the formulae used to perform the calculations. Though cancelled, the AC is still available online (I have a print copy).



EDIT: See the comment below from @voretaq7 referencing the excellent "Aircraft Weight &amp; Balance Handbook" quoted here: 


  In addition to AC 65-9A the current Aircraft Weight &amp; Balance Handbook
  (FAA-H-8083-1A - available here) contains more than you ever wanted to
  know about computing weight &amp; balance.


This is currently the direct link to the PDF of that document.