How does some matter prevent other matter from reaching a lower potential energy state? Before answering the question, keep in mind that I am a second year Biology student, with no experience in studying Physics and a very basic understanding of Mathematics. However, I have some experience studying introductory Chemistry.

An example of matter preventing other matter from reaching a lower potential energy state, is a table carrying an object, preventing it from falling due to gravity. However, I find it hard to understand what energy interactions are occurring between the matter involved.
After Marcello Fonda kindly explained the terminology and concepts relevant to my question, I have refined my question and clarified my understanding of the subject.
I understand that energy is,

a measure of an object's ability to interact with other objects, changing their state of motion

and that,

Holding an object in place maintains its potential for interaction
(energy), it doesn't change it.

Initially, I suggested that the force present keeping the weight of the object (which I assume is a product of the amount of matter in the object and the 'strength' of gravity) at a higher potential energy, was possibly due to the repulsive and attractive forces between the atoms constituting the table.
Marcello Fonda has agreed that the static force is caused by atomic repulsion. This makes me wonder if increasing the weight of the object on top of the atoms constituting the table, will cause them to be 'squeezed' or compressed, thereby increasing their energy state?
If so, where would the source of energy pressing the atoms together come from?
Side note: it would be nice to know the formal name for these 'barriers', or more specifically, the name for the matter preventing the other matter from falling to lower potential energy states.
 A: The concept of energy is a rather mysterious one. Feynman's explanation is probably one of the most beautiful, simple, and clear I've seen so far, you should try and check it for a good understanding.
The main point of the answer to your question, though, is that we like to define energy as being always related to the force used for moving objects, not for holding them in place. To keep an object still doesn't change its energy, although force may be needed: you can thus think of energy as a measure of an object's ability to interact with other objects, changing their state of motion, for example by colliding with them or by pulling or pushing them around. Holding an object in place maintains its potential for interaction (energy), it doesn't change it. If you release the object by removing the table, you can turn that potential for interaction into real interaction, for example by tying your object to a pendulum clock and have it move the dials while losing height (by the way, this is sometimes called a gravity battery).
A change in energy is called work, so for those forces which cause no change in an object's energy we say that they do no work. They are often called ideal constraint forces, so the table in this context would make for an ideal constraint.
The main reason all this stuff sounds counterintuitive is that we as animals consume energy even for static holds, but that is due to the fact that our muscles work by continual micro-contractions, not by a single hold. A static hold is never truly static for our muscles, rather is a very fast cycle of pull-and-release type oscillations (you might be aware of the actual mechanism) which actually does work at a microscopic level (it takes the energy from ATP bonds), though not at a macroscopic one. A table doesn't experience this kind of fatigue, since atomic repulsion is rather static in nature.
EDIT to address further question:
Increasing the weight will make the table bend and squeeze. Its atoms will then be at a higher energy state because of the stretching and squeezing of their bonds (which can be reasonably accurately be thought of as acting like microscopical springs). This increase of energy will come at the expense of the energy of the object, which will fall a little distance by bending the table downwards.
What we expect is that the bending will eventually stop. This can be explained in terms of energy by the fact that, as you bend the table, the energy required to further bend it increases. You reach a point where lowering the object yields less energy than is required to bend the table, so the descent stops.
If no energy is dispersed in the process, lifting the weight will let the table go back to its initial, straight position. If there is energy dispersion (bonds are broken), then the table won't be able to get back to its initial position. If the dispersion is slow, the table will get saggy. If it is fast, the table will crack and break.
Note that all this is regardless of the table's height: unless the object is tied to the ground with some sort of spring, raising the table and the object together (like carrying them to the second floor) doesn't stress the table more. Otherwise we couldn't have desks in skyscrapers.
A: A very partial answer to address a misconception, since this is a multi-part question. You wrote:

This makes me wonder if increasing the weight of the object on top of the atoms constituting the table, will cause them to be 'squeezed' or
compressed, thereby increasing their energy state? If so, where would
the source of energy pressing the atoms together come from?

Atoms and solids and gravitational fields are inconveniently complicated, so I'll start by replacing this complicated system with a very simple system.
Consider Alice and Billy playing on a see-saw. When Alice's separation from the planet decreases (the gravitational field does work on Alice), Alice doesn't need a rocket to keep her from smashing painfully into the ground - she just needs somewhere to put the energy that she's getting from the gravitational field. Alice puts the energy into the separation between Billy and the planet: when Alice goes down, Billy goes up.
Back to the weight on the table. If the table and the weight were both perfectly rigid (no amount of force can make either deform without breaking) then, if we started the weight resting on the table, the gravitational field wouldn't do any work because the weight wouldn't move, so the weight wouldn't need anywhere to put the energy - the table could be perfectly fragile in addition to being perfectly rigid, and nothing would happen. That, of course, is not how reality works: perfect rigidity is a frequently useful counterfactual, not a property of real objects made of atoms connected by inter- and intramolecular forces.
So we have a deformable weight on a deformable table, and we apply the force of gravity, which starts deforming the weight and the table. This allows the center of mass of the weight to go down: the gravitational field has done work on the weight.
Just like Alice and Billy, we don't need a rocket expending energy to push in the opposite direction to keep the weight from falling through the table. We just need somewhere to put the energy. For the weight and the table, the place where the energy can go is into the intramolecular force fields holding the solid together. Much like a spring (or, a bit closer to reality, like an incredibly complex system of countless, constantly jiggling, tiny interconnected springs), the solid has a higher energy state when it is deformed by an external force than when it is resting naturally.
The energy to cause the deformation comes from the gravitational field, which is a mathematical representation of the energy associated with the configuration of centers-of-mass in the system.
Stopping the deformation does not cost extra energy (unless you do it with a rocket). It demands a repository to put the energy into. The energy that comes from the configuration of centers-of-mass in the system goes into the configuration of interacting molecules in the system. If the configuration cannot change in such a way as to hold that much energy in that amount of time, either the table or the weight must break.
