Energy:  Vigor or power in action.  The capacity for accomplishment or action.

Power:  The ability or capacity to act or perform effectively.  Strength or force exerted or capable of being exerted; might.  The ability or official capacity to exercise control; authority.

Work:  Physical or mental effort or activity directed toward the production or accomplishment of something; toil; labor. 
 

Energy:  The work that a physical system is capable of doing in changing from its actual state to a specified reference state, the total including, in general, contributions of potential energy, kinetic energy, and rest energy. 

Power: The rate at which work is done, mathematically expressed as the first derivative of work with respect to time…

Work:  The transfer of energy from one physical system to another; especially, the transfer of energy to a body by the application of a force, calculated as the line integral between any two points of the scalar product of the force and the body’s displacement along the path over which the integral is taken.
 

Definitions have rules.  One person might want to define “vegetable” with “A vegetable is a potato.”   A second person says, “A vegetable is a carrot.”  And so they argue.  Which is right?  Is it potato or carrot?

Absurd, of course.

But defining energy with “Energy is the capacity to do work” makes the same error (an inverted implication: see box at right), and it is just as absurd.  The big difference is that almost everyone sees the vegetable absurdity and almost no one sees the energy absurdity.  Both make a certain kind of oversimplification.

Look out for oversimplifications by which something multiply related is seen as some single thing.  More than one thing can be a vegetable.  Capacity to do work involves more than just energy—and thermodynamics was developed in the 19th century to sort out the tangle of multiple influences.

If you have energy, you may or may not have capacity to do work, but if you have capacity to do work, you must have energy.  That is, capacity to do work implies energy, but not the other way around.

Similarly, potato implies vegetable, but not the other way around.

The simple logical relationship of implication is easy to see when it’s about concrete things.  When it’s about abstractions it becomes “simple but subtle,” a description often used for physics in general.

Some version of the above dictionary scientific definition for energy (in red) is used in about half of physics textbooks, but it is logically flawed by being an “Energy is the capacity to do work” definition. 

Most textbooks that avoid this error also avoid giving any definition of energy. 

One physics textbook that stands out, by avoiding the logical error and still addressing the question “What is energy?,” is The Feynman Lectures on Physics.  Feynman tells us that “...in physics today, we have no knowledge of what energy is.”  He goes on to say that we know how to calculate its value for a great variety of situations, but beyond that it’s just an abstract thing which has only one really important property.  If we add up all the values before something happens and then add them up after it happens the two values will be exactly the same.  (We must be sure to include every object affected.)  This is the law of conservation of energy.  Note that the "every object affected" criterion is a better way of expressing the requirement usually referred to as "closed system."
Thermodynamics is about transfer of energy.  The scientific definition of  “work” points to one kind of energy transfer.  There is another kind: heat.  And it is here that the many parameters of the real world enter and generate the subtlety that blinds us to the reality.

Heat is a statistical (“stochastic” if you prefer) transfer of energy.  If the needed insight into things statistical were widespread in society, there would be no state lotteries or gambling casinos.  Heat is like the flow of money at the gambling tables: a lot of randomness with a definite trend asserting itself with time and large numbers of transactions.  (Work is like the money that flows from the casino office to the bank: an organized, predictable flow all in one direction.) Temperature is the parameter that determines the trend.  And entropy* is the parameter that renders energy unavailable for doing work.  To understand the colloquial meanings of “energy” we must study entropy—and entropy is central to what we call “waste” products. 

Discovery of what energy is eventually turned out to be the discovery of thermodynamics: the discovery of a web of relationships between many parameters, including such entities as energy, pressure, temperature, volume, entropy...   These many parameters were found to have a subtle intricacy between each other, understandable to us humans only in mathematical language.

To see this web of relationships, human beings had to develop insight into unfamiliar subtleties: relevance compared with irrelevance, statistical relationships, subtle contradictions, and such details as the distinction between necessity and sufficiency and between equivalence and mutual exclusion.  These are very subtle insights, often obscured by oversimplified "seeing," and even by wishful thinking.

The very serious problem tripped over by Feynman as he read the K-12 science textbooks is the pervasive, persistent, and pernicious nature of our pre-scientific oversimplifications.  The problem is found throughout science and mathematics...wherever the science was not immediately "obvious" to human beings through our (limited) perceptions but was instead revealed to us only through deep thought.  Each of us who learns some simple piece of modern science must first heed and attend to the warning:  

"The process of science cannot be learned by reading, listening, memorizing, or problem-solving.  Effective learning requires active mental engagement."  (U of Wash)