We get this question A LOT on various public forums. People seem to want to know what it is, as if it is a "quantity" that should have just one basic and complete definition. It is one of those words in which we use very often and assume that we and everyone else know the meaning of.
However, what is used in the ordinary language is not necessarily the same thing as that used in physics. This is especially true when one considers that fact that everything in physics must have an underlying mathematical description. Add to that the complication that many things in physics get more complicated and more complex as one tries to define something in the most general form that covers ALL areas of physics from classical mechanics to elementary particles.
This article, written by a retired physics professor, tries valiantly to define what energy is to the public as used in physics. I'm impressed that a local newspaper would carry such a thing. It is a useful starting point for a lot of people to differentiate between how that word is used in ordinary pedestrian language versus how it is used in physics. Now I know I'm being a bit nitpicky here, but the article became a bit confusing, and incomplete, towards the end.
I like how he clearly indicates right away that there are 3 different forms of energy. The first being the energy of motion (kinetic), the second being energy a particle has in a field (which is just potential energy). However, I'm confused to what the 3rd form of energy is since, unless I missed it, he didn't state it explicitly. The way it went, I'm guessing he is saying that it is thermal energy. However, if this is true, then it isn't really a different form of energy because in the article itself, he stated that heat is nothing more than kinetic energy. This, as we all know, is correct from classical thermodynamics.
So based on this article, there should only be two forms of energy. This is quite accurate, especially when we only consider classical mechanics. In solving the dynamics of a classical system, when we write the Hamiltonian or Lagrangian, we consider the T and V of the system, which are the kinetic energy and the potential energy respectively, in terms of generalized coordinates and generalized momentum. That is all that we need. This shows that those are the only two forms of energy that is the "source" of other energies, and the only forms of energy necessary to completely describe the system.
We similarly have the same thing in quantum mechanics where the two terms the Schrodinger equation are the kinetic energy and potential energy.
However, we can also argue that there IS a 3rd form of energy, that is being the conversion between mass and energy via the infamous Einstein equation. Because of the conversion between mass and energy, the more general relevant conservation law would have to include the sum of both mass and energy when there is a conversion from one to the other. So in this article, the 3rd form of energy should be the energy equivalent represented by a mass. Notice that I hesitate to say that mass and energy are "the same thing". The connection isn't that easy and we are still studying such a thing, especially in the origin of "mass".
One could also argue that photons are another form of energy. However, if one does not want to be to picky, one can easily argue that photons have "kinetic energy" via their momentum, so one doesn't have to make a new category for them.
As I've said earlier, as one tries to cover all the bases and try to be as general and complete as possible, it get more complex and things aren't as easy as they seem. However, I do think that this should cover what most of the general public would need to know on what "energy" is. If they want more, they'll just have to study physics! :)