Does energy exist, or is it just a relationship between other fundamental things? Could anyone tell me what energy really is? I searched for it, and some people said that energy doesn't exist physically and it is not fundamental, but it is a relationship between other fundamental things, and there is not energy by itself, so it should be related to something else.
So could anyone help me understand it?
 A: "What energy is" is a philosophical question.  It turns out its impossible for science to talk about what "reality" is like, other than to say that science forms models which have an "energy" term in them and they seem to be pretty good predictors.  If you're interested in that line of reasoning, I highly recommend looking into the philosophy of science.
However, we can find energy as a meaningful thing in our models.  One of the foremost ways of modeling our reality for scientific purposes is in the idea of "action."  The idea of action is formed from this question:
Given a path that a system may take from state 1 to state 2, what path does it take?  In other words, if someone throws a ball (state 1) and later someone catches it (state 2), what did it do along the way?
What we have noticed through decades (and even centuries) of observation is that you can phrase this as a minimization problem (more formally, a stationary problem, which is a wider concept, but minimization is easier to think about).  We noticed that you can define a function for a system, called the Lagrangian, such that if you integrate it across the entire path the system takes through time, it's at a minimum (at a stationary point, in the complete version).  This integration across all time is called the "action" for the path taken.  Interestingly, this function works for all configurations the problem might take on.  You can find an action describing that ball flying through the air which works not only for your thrower and catcher as they are, but a thrower and catcher anywhere on the field!
This is a very abstract concept, and it's okay if it doesn't make 100% sense when you first work with it.  But what makes it important was that we came to this concept of Action without invoking forces or energy, or any of those other terms.  We just pointed out that the paths objects take tend to be the one which minimizes action across the entire path.  Or, more generally, we determined that you could find a Lagrangian for which the "correct" path is always found by solving this optimization problem that minimizes the action.  Actually figuring out a Lagrangian function which does this is another matter, what matters is that one exists!
Now should you accept this declaration that there always exists a Lagrangian function such that the correct path of objects is always found by minimizing the action?  Perhaps not.  Don't take my word for it.  Science is an empirical art, not a purely mathematical art.  It's the observation of scientists over the centuries that say "We can always describe the motion of particles this way!"
Now once you have this, we then can invoke one of the most powerful mathematical formalisms in all of physics: Nother's Theorem.  This theorem shows that if you have a system which is described by this optimization problem, this minimization of action, and it has a continuous symmetry, then there is some conserved value.  This is neat because it takes some very abstract mathematical concepts, like continuous symmetries and action, and ties it directly to the idea of conserved values.
One continuous symmetry that's very important to physicists is time symmetry.  Basically that says that the laws of physics don't change over time.  We're only looking at laws that stay constant forever, from the big bang to however we end.  The laws of physics being the same at all times is formally termed as "time symmetry."  If you do something at one time, or do it 5 seconds later, the laws of physics will be the same in both cases.
This continuous time symmetry must have an associated conserved value, by Nother's theorem.  We call that conserved value "energy."  And if you actually go through all the fancy Calculus of Variations, you find that the thing that we conserve when we conserve energy is precisely what we told you was "energy" all along.
So, down in the weeds, that's the neat nature of energy.  Energy is the thing that must be conserved if the laws of physics are immutable over time.
A: Well, maybe it's not important to know what energy is, as much as it is to know what energy can do. The typical definition of energy is that energy is the capacity to do work. If something has energy, it has the capacity to move things, lift things, heat things, and so forth. The other thing that is important to know is that energy can never be "created" or "destroyed". In other words, total energy is always conserved. It simply morphs into different forms as it is transferred between things in the form of either heat or work.
Hope this helps.
A: I think term energy has slightly different meanings in different branches of physics. I prefer to think of it as the quantity that is conserved due to time-invariance of the equations of motion, i.e. my notion of energy is related to the Largangian mechanics (https://en.wikipedia.org/wiki/Lagrangian_mechanics), and energy is basically the Hamiltonian.
A: Energy doesn't exist itself, it's just a mathematical representation to make a relationship between the capacity of several objects to develop work. I can make an analogy. Energy is like money, doesn't have value itself, it only represents the value of things (work, savings, estates, ...)
