Conceptually, I've always understood entropy to be a statistical idea. For example, if you have a vacuum inside of a box and you place a handful of gas atoms on one side, the molecules have a higher statistical probability to spread out, versus remain in one concentrated spot. Therefore, on average, they will spread around and entropy is increased.
Of course there's a more elaborate definition involving macrostates and microstates, where entropy is greater when a macrostate has more microstates. However, in my understanding, the idea is still the same: increasing entropy is really just a statistical likely-hood that will hold true on average.
There are many formulas that relate entropy to energy. For example, Gibbs Free Energy: $$ΔG = ΔH - TΔS$$
Now, before I explain my confusion, I'd like to share an example from my course: hydrogen bonding (non-covalent interactions... like electrostatic interaction).
We call the formation of hydrogen bonds as being "stabilizing" - that is, they lower the overall energy $ΔG$. However, when hydrogen bonds form, the $ΔS$ actually decreases; there is more order when molecules align to form H-bonds. But this entropy cost is offset by a greater decrease in enthalpy, bringing the overall $ΔG$ down (so it's negative), and therefore making the bond "stabilizing".
That's where my confusion begins. I'm having a difficult time understanding how we can relate the idea of entropy to energy. A paper that my professor once showed me makes it clear that entropy isn't a measure of energy density/distribution... rather, it's more of a statistical observation.
So, assuming that entropy is a statistical idea (which is where I think my gap in understanding may be), how does forming a hydrogen bond, or being ordered in general, "destabilize"/increase the energy? Why does being ordered - which is simply just a small probability event actually occurring - impact the energy of a system? It's not like some outside force is inputting energy into the system to align and order the molecules... so how is entropy, a statistical idea, playing a role in the energy?