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In elementary physics courses one is taught that energy is a measure of an object's ability to do work (this in itself seems a little flaky as how does one then define exactly what "work" is, other than stating that when a force acting on an system causes a displacement of that system, then the force is said to have done work on the system, or resorting to a tautological statement that describes it in terms of energy?!). Then in more advanced courses this description usually becomes more abstract in as such that energy is a conserved quantity attributed to a physical system, or an emergent quantity as a natural consequence of time translation invariance of a physical system (via Noether's theorem).

My question is, though, what exactly is energy? Is it possible to give a qualitative description of what it is a measure of?

I have read through the answers given in What's the real fundamental definition of energy? and although they are somewhat useful, I feel that they don't really answer my question in terms of whether we can give a qualitative statement of what energy is, by this I mean, for example, velocity is a measure of the rate of change in position of an object per unit of time, is there a similar statement that can be ascribed to energy (or momentum for that matter)?

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Will
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I would argue that energy isn't a measure of anything at all - it is a thing. That's in the same way that matter isn't a measure of anything either. Given that E = mc², they are arguably different forms of the same thing.

You can describe energy in terms of what it can do - pushing things along in the case of doing work - but energy isn't a measurement like velocity is. You can also measure how much of it you have, in the same way that you can measure the mass of a lump of matter.

If energy is a thing, then conservation of energy intuitively makes sense. You can't make matter just disappear. You can move it or turn it into a different form, but it's still there. The same goes for energy.

Aside: the English language has distinct words for "matter" (the thing) and "mass" (the amount of the thing). The terminology gets a bit woolly when describing energy.

Simon B
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Energy is like money - what you have in your pocket is only one of its many forms. Both are abstract ideas convertible between many forms.