Is there an interpretation of (potential) energy? As the title says, is there some sort of physical interpretation of potential energy? While I understand that kinetic energy is energy that something has by virtue of its movement (and this is obvious from the formula $\,\mathrm{KE} =(1/2)\, m\,v^{2}\,$), but what about potential energy? A lot of sources say that potential energy is energy that an object has due to "its position relative to others" but what does that really mean? Does there need to be a force acting on an object for potential energy to exist?
 A: I disagree with ison. Potential energy is not just something which could give rise to energy. It is energy, which is stored in some degrees of freedom other than motion. E.g. 


*

*If you lift a weight 1m above ground (on earth) then you need to perform work. This work is not stored as kinetic energy, but as potential energy $E_{pot} \approx m g h$. 

*If we stretch a spring, we need to perform work. Again, the work is not stored as kinetic energy, but as potential energy $E_{pot} = \frac{1}{2}k (\Delta x)^2$, where $k$ is the spring constant and $\Delta x$ is the distance from the so called equilibrium point.

*If we separate a positive and a negative charge ... 


In all these examples it's the relative position objects, which give rise to the potential energy:


*

*In the first example it's the gravitational force against which we lift a mass. Work is defined as $\Delta W = F \cdot \Delta s$, so we perform work by lifting the mass. This energy is stored as potential energy.

*In the second example it's the spring force against which we stretch. 

*In the third example, we would perform work against the electrostatic force.

A: Potential energy is a type of energy which is in fact related to "relative positions" but more important is in a one-to-one correspondence with conservative forces.
Let us consider a particle starting at point $A$, finishing at $B$ and subjected to a force $\vec F$ which can depend on position. It is quite important to know how the work $W$ done by this force depend on the trajectory from $A$ to $B$. If we want to save energy we should choose the path giving the least work. It happens that for some forces, called conservatives, the work does not depend on the path so it can only depend on the initial and final points. Given this conservative force we define a quantity $U$, called potential energy, which takes real values over space and such that the work of $\vec F$ from $A$ to $B$ equals the negative of the difference between the values of $U$ between $A$ and $B$, i.e., 
$$W=-\Delta U.$$
Since what matters is only the difference $\Delta U$, there is an ambiguity about the zero of potential and you are free to associate any value $U(A)$ to a given point $A$.
As you can see, if there is no force at all, there is no work and therefore $\Delta U=0$ for any points $A$ and $B$. This means that potential energy can indeed exist but it is constant over space (of course you can take this constant to be zero). Gravitational force, electrostatic force, Hooke Law force are examples of conservative forces. Associate to any of these forces there is a potential energy, such that $W=-\Delta U$. On the other hand, dissipative forces such as friction are not conservative and therefore are not associated to a potential energy.
A: It's exactly what it sounds like. "Potential"+"energy". It's the energy something doesn't actually have yet, but has the potential to gain based on its position. If something is on the ground it has no potential to gain energy. If it's high up in the air then it has the potential to gain a lot of energy as it falls.
Note, however, that this is really just one interpretation to give intuitive insight in basic physics. In actuality there is no real reason to say that potential energy doesn't count as a legitimate form of energy, and many areas of physics are more easily understood if you treat it as such.
A: How hard is it to keep a thing the way it is? How much energy can you get if you allow it to do something else?
Really intuitively drop a brick: it hits the ground with some kinetic energy, that energy came from the potential energy the brick had from not being on the floor.
Potential energy is mostly useful in balancing conservation of energy. If there is energy in a falling brick it had to already be in the system.
When you hold a brick up you feel its weight (mass * gravity). You could let the brick exert that force on something other than you over a distance of how high you can let it down. For gravitational potential energy the equation is Energy = mass * gravity * height. If you cut a hole in the floor you could let it down farther, and you'd need to re-base your height measurement for how far the brick can now drop.
