The aircraft is weighed with all the equipment necessary to fly minus the fuel and crew.
People smarter then me determine an aircrafts desired CG.
To compute weight and balance one needs a weight and moments.
Moments are drived from a given weight x the arm where the weight is. Arm being the location of the weight measured in inches from the reference datum line usually 30" or so in front of the nose. All measurements are taken from that spot.
Take my kingair 200 for example. It's weight with crew and gear minus fuel is 8,906 pounds with 16,514 moments.
(in this example moments are simplified by a 100 in larger airplanes like the C-130 moments are simplified by 1000)
So as it sits right now the CG is 185.4 inches from the RDL. Those people smarter then me have determined that a safe CG range at 8,906 pouns is 181.0 to 196.4.
Now when you add fuel you add the fuel weight and the moments it creates ( i don't have my chart handy so these numbers may not be all that accurate but enough for a lesson) If I had 2625 pounds of fuel that will give me a moment of 4848.
So now the airplane weighs 11531 with a moment of 21362 so now the plane has a CG of 185.3 Allowable range at this weight is roughly 181.3-196.4 ( i don't have my chart so bear with me)
Now add some passengers to the mix.
We'll say 2. 1 is going to sit at fusalage station 220(also known as the arm) the other one sits at 262.
We use a standard weight of 200 pounds per person and it keeps us pretty safe
So passenger one is 200 pounds and creates 440 moments passenger 2 is 200 pounds and creates 524 moments.
Now if we add in the 400 pounds and the 964 moments to our operating weight of 11531/21362 we get
11931 pounds and 22326 we get a cg of 187.1 the given range for this is roughly 182-196.4
So by knowing the weight and where it is placed on the airplane you do a little math and you find where the balance is. The more weight you add to the aft the farther aft that CG will go.
This is what I do every day. For every flight you also have to figure out the landing CG based on fuel on board. But if you don't do any air drops or tanking in flight if your with in balance at take off and your with in balance at zero fuel landing will be good.
There are other parameters to watch out for as well. Such as max weights, wing fuel limiting etc. Wing fuel limiting is basically a minum fuel load for a given cargo weight. You can't stuff a plane full of cargo and have a very little amount of fuel. If you exceed those limits then there is not enough fuel in the wings to hold the wing down in flight and will cause it over stress and then you could end up with the wing off light.
If you have any questions shoot em my way.
there is computer programs out there that will do all the math for you but I have not been privy to any of them and do this by hand for every leg. It gets to be tedious when you know from pure experience that if I put 2 passengers on this plane they could sit just about anywhere and I'll be well with in limits.
Some planes need to know where the CG is so the pilots can set the proper amount of elevator trim for take off. I have not crewed on any of those planes. So in my case all it tells the pilot is your either gonna have a heavy nose, a heavy tail or fairly neuteral feel on rotate. You can tell the difference from one extreme to the other but not to the point you have to worry about a tail strike if your with in limits.