Weight and balance! Very important, I have no experience with loading aircraft for scheduled passenger service but this is what I know...
For my final year design project at uni (remote controlled plane) I had to calculate the weight and balance; I did so by estimating my centre of lift at the quarter chord of the (straight) wing, I then lumped my various components into their own specific categories and calculated their weight based on volume and density of the materials, and then estimated their individual CofGs. Once I had the CofG and weight of each 'module' or component, such as engine, centre fuselage, tail, battery, fuel tank, landing gear etc, I then worked out the distance between each of the CofGs and the quarter chord which is where I wanted the whole aircraft CofG to be. With the distances worked out, I could then work out the moments of each by multiplying the weight by the distance. For moments behind the quarter chord you take the distance as a negative number, so that gives a negative moment, for moments in front of the quarter chord the distance is a positive number which gives a positive moment.
After calculating all the moments, with negative for behind the target CofG and positive for in front, you add them all up. If the design is 'balanced', the sum of all the moments should come to zero or very close to it. If the number is greater than zero, you know your CofG is in front of your estimated centre of lift, if it's less than zero then your CofG is behind your estimated centre of lift.
You can create an excel spreadsheet to do all of the above very quickly, and then you can also make a pretty graph of the weight distribution of your aircraft.
I'm also learning to fly, and as a student pilot, you're taught a very similar process, although with a Cessna 172 pilots operating handbook there are lots of helpful tables and graphs to help you work out whether you'll be in balance much more quickly; no excel spreadsheets necessary, just a calculator.
And finally, in a real world design environment, at work (I'm an aircraft systems design engineer) if I make a design change which has any kind of weight impact I have to go talk to the 'weights engineers', who will then tell me if my design change is catastrophic or insignificant, hehe... Mostly they're happiest when the airplane is getting lighter
Seriously though, weight and balance all of a sudden gets more complicated when you have swept wings, variable flap settings, spoilers and fuel tank cross-feed to think about, so the weights engineers obviously go into more detail than I did with my remote controlled plane.
And finally, google turned up this-
It seems you can buy software to balance the aircraft for you, and I imagine most airlines have something similar.