Why Is Potential Energy Critical in Understanding Physical Systems?

[joebo]

F=-$$\nabla$$V
i cannot fully understand it ,though i can deduce it in some ways.

i also want to know why potential energy is so extensive and important.

 

[fluidistic]

F represent a vector field. More precisely, a conservative vector field. Hence it is the gradient of a function which we call the potential.
I hope it helps a very bit at least.