What Is Energy? Where did it come from?

Energy is any quantity - a number with the appropriate units (in the SI system, Joules) - that is conserved as the result of the fact that the laws of physics don't depend on the time when phenomena occur, i.e. as a consequence of the time-translational symmetry. This definition, linked to Emmy Noether's fundamental theorem, is the most universal among the accurate definitions of the concept of energy.

What is the "something"? One can say that it is a number with units, a dimensionful quantity. I can't tell you that energy is a potato or another material object because it is not (although, when stored in the gasoline or any "fixed" material, the amount of energy is proportional to the amount of the material). However, when I define something as a number, it is actually a much more accurate and rigorous definition than any definition that would include potatoes. Numbers are much more well-defined and rigorous than potatoes which is why all of physics is based on mathematics and not on cooking of potatoes.

Centuries ago, before people appreciated the fundamental role of maths in physics, they believed e.g. that the heat - a form of energy - was a material called the phlogiston. But, a long long time ago experiments were done to prove that such a picture was invalid. Einstein's $E=mc^2$ partly revived the idea - energy is equivalent to mass - but even the mass in this formula has to be viewed as a number rather than something that is made out of pieces that can be "touched".

Energy has many forms - terms contributing to the total energy - that are more "concrete" than the concept of energy itself. But the very strength of the concept of energy is that it is universal and not concrete: one may convert energy from one form to another. This multiplicity of forms doesn't make the concept of energy ill-defined in any sense.

Because of energy's relationship with time above, the abstract definition of energy - the Hamiltonian - is a concept that knows all about the evolution of the physical system in time (any physical system). This fact is particularly obvious in the case of quantum mechanics where the Hamiltonian enters the Schrödinger or Heisenberg equations of motion, being put equal to a time-derivative of the state (or operators).

The total energy is conserved but it is useful because despite the conservation of the total number, the energy can have many forms, depending on the context. Energy is useful and allows us to say something about the final state from the initial state even without solving the exact problem how the system looks at any moment in between.

Work is just a process in which energy is transformed from one form (e.g. energy stored in sugars and fats in muscles) to another form (furniture's potential energy when it's being brought to the 8th floor on the staircase). That's when "work" is meant as a qualitative concept. When it's a quantitative concept, it's the amount of energy that was transformed from one form to another; in practical applications, we usually mean that it was transformed from muscles or the electrical grid or a battery or another "storage" to a form of energy that is "useful" - but of course, these labels of being "useful" are not a part of physics, they are a part of the engineering or applications (our subjective appraisals).

I don't think the answer is trivially simple. I will try to give an explanation. In many problems of physics, what you are given is the initial and final states of the system. You don't know (or maybe no one does) what happens between these two states. Now there are quantities that you can measure before and after the system has undergone this change of state. The question is can you predict some of these quantities by knowing the others. Remember that we don't know the mechanism by which the system moves from these two states. But if you have something known as a conservation law, the problem becomes simple. (By saying that a quantity is conserved we mean that it doesn't change throughout some process). Suppose you have some magic function involving the quantities, which gives the same value no matter what the state of the system is, then you are done. The value of the function we call energy. And since its value doesn't change between these two states we say that its conserved.

This excerpt is from Feynman Lectures:

There is a fact, or if you wish, a law, governing all natural phenomena that are known to date. There is no known exception to this law—it is exact so far as we know. The law is called the conservation of energy. It states that there is a certain quantity, which we call energy, that does not change in the manifold changes which nature undergoes. That is a most abstract idea, because it is a mathematical principle; it says that there is a numerical quantity which does not change when something happens. It is not a description of a mechanism, or any- thing concrete; it is just a strange fact that we can calculate some number and when we finish watching nature go through her tricks and calculate the number again, it is the same. (Something like the bishop on a red square, and after a number of moves—details unknown—it is still on some red square. It is a law of this nature.)

To understand what energy is, it is necessary to understand the concept of work.

Work is defined as the action of a force over a path.

$$ W=\vec{F}\cdot\vec{d}$$

What does this means? It describes how "exerting" or "draining" a particular action is. For example, imagine lifting a shopping bag of mass $10\ \mathrm{kg}$ vertically by $1\ \mathrm m$. This takes work, and exactly the following amount, given by the weight of the bag times the distance.

$$W= \vec{F}\cdot\vec{d} = Fd\cos{0}=mgd=10\ \mathrm{kg}\times9.8\ \mathrm{m\ s^{-2}}\times1\ \mathrm m=98\ \mathrm J$$

Energy is classically defined as the capacity of a physical system to do work, or in other words: as you perform work, you exchange energy for some physical effect by doing work. Or in other terms again, by exerting a force over a distance you convert energy into work.

In our example, you need to use some form of energy to lift the shopping bag. The quantity you need is exactly the amount of work we calculated.

What happens to this work? It's converted to energy again – to gravitational potential energy:

$$U_\text{final} = U_\text{initial} + W$$

or

$$\Delta U = U_\text{final} - U_\text{initial} = W = mgd$$

which is the classical definition of gravitational potential energy.

So in practice – we never see or measure energy directly. When energy changes form, it is called work, which we can measure. So work, in a way, is a "transport" concept for energy. Energy, on the other hand, is like a "reservoir" of work in potential.

Why is energy a useful quantity? After all, work seems to be a more "fundamental" quantity from an experimental point of view.

The answer to this lies in the conservation law of energy. Work in itself describes a change in energy, so it's not a conserved quantity in itself unless you embed it in the more general concept of energy, which is conserved.

In fact, we can derive large swaths of classical mechanics using conservation of energy as a prime principle, together with the principle of least action.

Caveats

In more advanced theories, conservation of energy is a much more complicated matter and does not apply as simply as in the classical sense. For example in SR, energy can be converted to apparent mass and vice versa.

There are also very interesting mathematical properties of potential energy and its relation to forces and especially fields of forces. These explanations, though are way more abstract and mathematical – I assume you want an intuitive, instinctual explanation of what energy is.

If you are looking for the former please see this question.