Lectures on Physics has been derived from Benjamin Crowell's Light and Matter series of free introductory textbooks on physics. See the editorial for more information.... 
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Gravitational Potential EnergyAll the vital points about potential energy can be made by focusing on the example of gravitational potential energy. For simplicity, we treat only vertical motion, and motion close to the surface of the earth, where the gravitational force is nearly constant. (The generalization to the three dimensions and varying forces is more easily accomplished using the concept of work, which is the subject the next chapter.)
To find an equation for gravitational PE, we examine the case of free fall, in which energy is transformed between kinetic energy and gravitational PE. Whatever energy is lost in one form is gained in an equal amount in the other form, so using the notation ΔKE to stand for KEf  KEi and a similar notation for PE, we have
It will be convenient to refer to the object as falling, so that PE is being changed into KE, but the math applies equally well to an object slowing down on its way up. We know an equation for kinetic energy,
so if we can relate v to height, y, we will be able to relate ΔPE to y, which would tell us what we want to know about potential energy. The y component of the velocity can be connected to the height via the constant acceleration equation
and Newton's second law provides the acceleration,
in terms of the gravitational force. The algebra is simple because both equation [2] and equation [3] have velocity to the second power. Equation [2] can be solved for v^{2} to give v^{2 }= 2KE/m, and substituting this into equation [3], we find
Making use of equations [1] and [4] gives the simple result ΔPEgrav = FΔy . [change in gravitational PE resulting from a change in height Δy; F is the gravitational force on the object, i.e., its weight; valid only near the surface of the earth, where F is constant]
It may be dismaying to note how many minus signs had to be handled correctly even in this relatively simple example: a total of four. Rather than depending on yourself to avoid any mistakes with signs, it is better to check whether the final result make sense physically. If it doesn't, just reverse the sign. Although the equation for gravitational potential energy was derived by imagining a situation where it was transformed into kinetic energy, the equation can be used in any context, because all the types of energy are freely convertible into each other.
Here are some questions and answers about the interpretation of the equation ΔPEgrav = FΔy for gravitational potential energy. Question: In a nutshell, why is there a minus sign in the equation? Answer: It is because we increase the PE by moving the object in the opposite direction compared to the gravitational force. Question: Why do we only get an equation for the change in potential energy? Don't I really want an equation for the potential energy itself? Answer: No, you really don't. This relates to a basic fact about potential energy, which is that it is not a well defined quantity in the absolute sense. Only changes in potential energy are unambiguously defined. If you and I both observe a rock falling, and agree that it deposits 10 J of energy in the dirt when it hits, then we will be forced to agree that the 10 J of KE must have come from a loss of 10 joules of PE. But I might claim that it started with 37 J of PE and ended with 27, while you might swear just as truthfully that it had 109 J initially and 99 at the end. It is possible to pick some specific height as a reference level and say that the PE is zero there, but it's easier and safer just to work with changes in PE and avoid absolute PE altogether. Question: You referred to potential energy as the energy that two objects have because of their distance from each other. If a rock falls, the object is the rock. Where's the other object? Answer: Newton's third law guarantees that there will always be two objects. The other object is the planet earth. Question: If the other object is the earth, are we talking about the distance from the rock to the center of the earth or the distance from the rock to the surface of the earth? Answer: It doesn't matter. All that matters is the change in distance, Δy, not y. Measuring from the earth's center or its surface are just two equally valid choices of a reference point for defining absolute PE. Question: Which object contains the PE, the rock or the earth? Answer: We may refer casually to the PE of the rock, but technically the PE is a relationship between the earth and the rock, and we should refer to the earth and the rock together as possessing the PE. Question: How would this be any different for a force other than gravity? Answer: It wouldn't. The result was derived under the assumption of constant force, but the result would be valid for any other situation where two objects interacted through a constant force. Gravity is unusual, however, in that the gravitational force on an object is so nearly constant under ordinary conditions. The magnetic force between a magnet and a refrigerator, on the other hand, changes drastically with distance. The math is a little more complex for a varying force, but the concepts are the same. Question: Suppose a pencil is balanced on its tip and then falls over. The pencil is simultaneously changing its height and rotating, so the height change is different for different parts of the object. The bottom of the pencil doesn't lose any height at all. What do you do in this situation? Answer: The general philosophy of energy is that an object's energy is found by adding up the energy of every little part of it. You could thus add up the changes in potential energy of all the little parts of the pencil to find the total change in potential energy. Luckily there's an easier way! The derivation of the equation for gravitational potential energy used Newton's second law, which deals with the acceleration of the object's center of mass (i.e., its balance point). If you just define Δy as the height change of the center of mass, everything works out. A huge Ferris wheel can be rotated without putting in or taking out any PE, because its center of mass is staying at the same height.
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