How does the center of gravity work? In free body diagrams, such as a beam attached horizontally to a wall, $F_g$ is always shown acting on the center of gravity of an object.  
My question - is this the case in real life, where gravity only acts on this point of the object? Or is gravity acting on all parts of the object, but that point is at the exact center of all the force?
 A: Gravity (treated as homogenous) is acting the same on all parts of the object, but if the object is rigid, internal forces allow the simplification that the centre of mass is where all the force acts.
Torque:
There is equal mass on both sides of the centre of mass so there is no net torque about it. If the pivot is at the centre of mass, the object will not turn, it will balance.
A: Gravity pulls in every single particle in the object.
This means that gravity causes a torque at every single particle around some other point.


*

*If you look at a point to the left, then all those torques sum up to a net torque which pulls clockwise.

*If you look at a point to the right, then they sum up to a net torque pulling counter-clockwise.

*If you look at a point in between, then some torques pull clockwise and some counter-clockwise. 


At some special point, the clock-wise and counter-clockwise torques cancel exactly out.
Since we can see that things do not start to rotate when they fall, we must assume gravity when averaged out to be pulling in this point. If we averaged it out to pull in any other point, there would be a net torque and the object would spin - and we know/see that this does not happen.
So, let's call this point the centre of gravity, while keeping in mind that we can average out gravity to pulling in this point as a simpler model to work with. 
A: Simplifications to mechanics problems are often made with respect to the COG because it is the average location of all the mass in an object.  For example, picture a half-filled plastic water bottle on a 45-degree angle.  The weight of this object at one end is very different from the other end, and so if you were to translate this object across, let's say the xy-plane while keeping the angle intact, it would therefore only make sense to calculate where the average location of where the mass in the water bottle is located, and just use that point in your calculation because it is one, simple calculation as opposed to a much large calculation.  Here is a helpful link about COG if you are interested in it.
