Weight & Balance calculations are important, because you need to keep them within acceptable limits. All aircraft have a maximum weight and CG range which should not be exceeded for safety and performance reasons. This information can be obtained from the Type Certificate Data Sheet, or actual POH from the aircraft. In real life, you will use the ACTUAL weight and balance information from the SPECIFIC aircraft you are using. Remember that an overloaded, or an improperly loaded aircraft will exhibit erratic performance and behaviour which is dangerous and illegal.
To make a simple calculation, assuming we are talking about a small light aeroplane such as a Cessna 150 for example, you need to add up all the weights.
The weights are multiplied by the "arm" which is the horizontal distance from the datum (reference point) to the centre of whichever item you are measuring. This will give you what's called the "moment" of the item. Some moments will be positive, while others will be negative, so you have to be careful on this.
At the end, you add up all the moments, and add up all the weights and divide (M/W) to get the centre of gravity.
Of course, before you can do all this, you have to go through some compromising steps. For example, you might have to adjust your fuel load, or take less baggage.
To balance the load, assuming you need to bring the airplane back to normal limits, you can use this formula:
wt= weight to be moved
WT= gross weight of aircraft
d=distance to move CG
D=distance to move weight item
You then check this against charts to make sure you are within the "envelope" of whatever category you are (normal, utility).
Finally, make sure your cargo is strapped down properly, because accidents have occurred where the cargo shifted and caused control and stability problems.