What do we mean when we talk about Gibbs Free Energy? Before I start, I'm aware that this question may be better suited on the Chemistry or Biology site, but it's my belief that physicists are more likely to have a clear understanding on what certain terms mean, so by all means move the question if you feel like it will get a better response elsewhere.
Okay.
In Chemistry we learn about this thing called the Gibbs Free Energy (which I understand is borrowed from Thermodynamics). It's pretty simple. $\Delta G < 0$, and the reaction is spontaneous. $\Delta G > 0$, and the reaction is not spontaneous. 
Other terms in the equation for Gibbs Free Energy are the total enthalpy change, which I interpret as the amount of energy that the system either takes in or releases, and also the temperature and total change in entropy.
Observe these graphs of an ambiguous 'Energy' plotted against the progress of the reaction:
http://upload.wikimedia.org/wikibooks/en/a/a6/Gibbs_free_energy.JPG
http://www.citruscollege.edu/lc/archive/biology/PublishingImages/c05_10.jpg
http://images.tutorvista.com/cms/images/101/exothermic-and-endothermic-reaction.png
The idea is the same. Some reactions take in 'Energy,' and the curve ends higher than where it began. Some reaction release 'Energy,' and the curve ends lower than it began. All reactions seem to require an 'Activation Energy' which prevents the reaction from occurring spontaneously.
Notice how the Y-axis has different names, such as Gibbs Free Energy, PE of molecules, and PE.  Is Gibbs Free Energy the same or different from PE? I'm not sure anymore. Also, in one graph, the change in Energy is portrayed as $\Delta G$, so a decrease implies spontaniety, and increase implies nonspontaniety. Yet both require an activation energy to proceed. 
One more thing to notice is the change in terms. In a Biology context, the terms are Endergonic and Exergonic. In Chemistry, it is Endothermic and Exothermic. Why different terms for the same idea?
I would very greatly appreciate an explanation for this, which has been bugging me for a while.
 A: Yep, borrowed from thermodynamics. For a system that is constrained under fixed pressure, fixed composition, and fixed temperature, the Gibbs free energy is minimized.
You're right to question the link between Gibbs free energy and potential energy (or kinetic energy or total energy ...). In fact we know that Gibbs free energy is defined by
$$ G = E - ST + VP $$
where $E$ is the total energy of the system, so obviously there is a difference between actual energy and free energy. A spontaneous reaction decreases $G$ but even then, $E$ may increase during this process (exothermic process), provided the increase in $S$ and/or decrease in $V$ is enough to compensate.
The idea of activation energy is not strictly thermodynamics. It has to do with considering intermediate states which are most definitely not in equilibrium, but which must be passed through to go from reactants to products. Those intermediate states are unfavourable since they are high energy, or low entropy, or high volume. Activation energy tries to express this as a high $G$ barrier, though it's good to keep in mind that activation energy is not really a quantitative concept.
