There have been several questions posted on the Stack Exchange along the lines of "What is energy?" Many them have been marked as duplicates by moderators. Among the surviving ones are: What is Energy made of?, What Is Energy? Where did it come from?, and Is energy really conserved?. Most of the answers posted to the questions at the above links have been written by advanced practitioners, and invoke mathematical abstractions, quantum and relativistic concepts, and Noether's thereom.
However, it is almost certain that most people seeking answers to these questions are students in introductory courses. I’ve been there. We learn some fundamental principles and are pleased by how they fit and reinforce each other as a whole. But we are often left somewhat confused about energy. It seems that energy must be fundamental too, and should interlock with the other principles, and yet the educational messages we receive about it appear to have contradictions that leave energy oddly disjointed from mechanics
Part of the problem is the notion that energy exists as a separate entity somewhere. That notion is stubborn baggage that has been attached to the energy concept since its inception in the time before science understood the atomic structure of matter, and the fundamental forces in nature. The sense that energy exists as an entity is reinforced in our education by such standard statements as “There is only a certain, unchanging amount of energy in the universe,” “Energy can neither be created nor destroyed,” “Energy changes forms,” etc. And while statements that energy is only a calculation can be found in many textbooks and other publications, those statements don’t penetrate effectively because they virtually always gloss over the next logical question—calculation of what, exactly?
The reification of energy in our educational messaging together with the lack of clear, explicit teaching of what energy actually is can cause confusion. This is especially true at the intersection of intuitive understanding and mathematical understanding. I remember puzzling over questions like, Is work energy? Why do we need the idea of work? If work can be negative, then energy can be negative? The kinetic energy theorem says that $F\cdot d$ can change kinetic energy. Why is that only true for kinetic energy? How is kinetic energy different than other energies?
I believe that this is unfortunate, and that many people are left believing that energy is a mystery beyond their grasps. Is it really? I don't think so. I decided to post this self-answered question. I anticipate that after reading the answer the reader's reaction may be: Well,this makes sense, but if it is correct then why isn’t energy taught and written about this way everywhere? All accepted ideas are hard to change in science-and therefore also in science education, and that's as it should be. But besides the long-standing traditional coverage of energy in science education, there is innate satisfaction in the view that energy exists as an entity, but is unknowable except as a mathematical abstraction. After all, this is true for most of the truths revealed in relativistic and quantum physics. And the mathematical model of energy does work perfectly well. The answer below may not be popular with those who have made their peace with energy in this way. But for those who long to have both the math and the phenomena the math describes to make sense, here comes the resolution, and its correctness should be self-evident.
Can energy be defined and explained in Newtonian terms like force and motion?