Yes CG calculation is a weighted sum (of weights, but lets call those mass to avoid confusion) with the weight for each dial being fixed. The result of the sum is divided by the total mass to get the CG.

This thing displays total mass on the left (that's just an ordinary non-weighted sum of all dials), and on the right is shows the result of dividing the weighted sum by the total mass.

Pretty much, it is computing a centre of gravity for the plane.

The CG needs to be in certain limits otherwise the stability of the plane is compromised.

You take the moment-arms of weights and sum them together and come up with mostly front-aft balance. You could have left-right imbalance as well obviously but this is mostly a function of fuel imbalance.