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Energy is a physical quantity that can be assigned to any system, and which was traditionally understood as the ability to do work.
Conservation of energy is one of the most fundamental empirical principles in science. As far as we know, energy can never be created or destroyed, though mass can be converted to energy and vice versa (which is what Einstein's famous E = m c2 means).
Popular and economic uses of the term
Today, "energy" commonly refers to energy in forms harnessed to fuel human activities, e.g. electricity, petroleum, natural gas, coal, nuclear, solar, wind.
The history of our understanding of energy is one of the best known examples of unification in physics. Over time kinetic energy, gravitational potential energy, heat, electromagnetic energy, sound intensity, and chemical energy all were described using a single measure, one which was conserved even as these types of energy changed into each other.
Perhaps the single most profound unification came from Einstein's theory of special relativity. In special relativity, the mass of an object contributes to its total energy, such that when the object is at rest, the energy contribution from that mass is described by the famous equation E=mc2. Subsequent experimental work has confirmed that mass is a form of energy, and that mass can be converted to and from other forms of energy.
In quantum mechanics, the Hamiltonian operator is closely related to both the energy of a system and the evolution of a system over time. This implies that conservation of energy is linked to the immutability of fundamental laws of physics.
Energy can be measured. Units for expressing quantities of energy are:
The Joule: the amount of energy expended in applying a force of one Newton over a distance of one meter.
The calorie: the amount of energy required to heat one gram of water by one degree celsius.
The British Thermal Unit (BTU): the amount of energy required to heat one Avoirdupois pound of water by one degree fahrenheit.
The Watt-Hour: Equal to 3600 Joules (a watt is one Joule per second).
Concept of work
The concept of work is what connects force and energy. A force is said to do work if, when acting on a body, there is a displacement of the body in the direction of the force, expressed via the formula
where the domain of integration is over the region in which the force is applied. For example, a force applied to a body lifting it to some height is said to do work on the body, changing the body's total energy. Work is a result of a dot product between two vectors, meaning that if the directions of the force and movement are perpendicular, no work is done by the force. For example, Gravity does no work on a body moving on a horizontal surface because the force is perpendicular to the movement. Since work is the change of a body's energy they have the same units. In the metric system, energy is measured in Joules.
Pseudo-scientific use of the term
Devotees of New Age beliefs use the term "energy" for almost anything for which they don't have a scientific explanation, including emotion ("I'm feeling some bad energy") and processes within the human body ("the flow of energy seems to be blocked"). They believe crystals, sounds, light, pyramids, and numerous other objects with which they associate magical properties, give off or transform some sort of "energy" that is traveling through the universal ether and affecting everyone with a positive or negative effect.
Everything is energy
Those who promote some form of pantheism or universal consciousness, as well as woomeisters, sometimes claim that science has proven that "everything is made of energy" and that this somehow explains the unity of all things (or psychic powers, or the "law of attraction") . While it's certainly true that any system has an energy associated with it, saying that "in science, everything is energy" is not even wrong. In the physical sciences, energy is a computable quantity, a number that can be associated with any system. Energy is not a substance any more than "mass" or "volume" are substances, and therefore nothing can be "made of" energy any more than it can be "made of" volume or "made of" mass. This technical sense of the word "energy" is different from the vernacular usage (which often treats light, sound, heat, and electricity as substances which are types of energy).
Regrettably, various forms of the "everything is energy" meme have been propagated in popular science, where it is sometimes used as a shorthand for the principle of mass-energy equivalence.
This is different from "String Theory" which suggest as a theory of everything that electrons and atomic particles are 1-dimensional "strings" of energy, and some related principles suggesting that possibly all strings are pieces of the same string (so that all things are connected but not, at all, in a hokey spiritual way). Though several string theories have exceptional cleverness and math behind them, there has been practically no suggestion for tests of falsifiability.
Energy as life/consciousness/goat
In keeping with New Ager's usage of "energy" to describe anything and everything, "spiritual" concepts such as God, the soul, consciousness, or life itself are often described as types of energy. Assuming that they mean anything like what real scientists mean by the term, it's total bunk to call any of those things "energy". For one, God and souls are physically undetectable, whereas energy is. How could we possibly know that God is a form of energy? For another, consciousness and life seem to be very specific, complex processes. Energy is a single scalar measurement of a system (one which can be easily calculated, by the way, without including any contribution from "consciousness"). To compare energy to these processes seems to be meaningless, like comparing the aesthetic appeal of a Picasso painting to its total weight.
Confusion with force
Exerting a force on an object does not necessarily result in the expenditure of energy. This is somewhat counterintuitive to us humans, whose muscles have to vibrate (and thus expend energy) every time they exert a force, even if the thing they're exerting a force against doesn't move.
Think of it this way: You have a Volkswagen beetle with a big heavy spring strapped to its front bumper. In front of you is a building with a bathroom scale strapped to the side, level with your front bumper. You slowly drive the Volkswagen into the building spring-first, compressing the spring tighter and tighter, until the bathroom scale you're driving it into reads 100 pounds. You then set the parking brake, turn off the engine, and walk away. The spring is still exerting a force of 100 pounds against that building, as evinced by the bathroom scale, but you're no longer expending any energy. That spring can keep on exerting 100 pounds of force against that building until it rusts away. Only if you release the parking brake, and let the spring push the Volkswagen backwards, will the energy stored in that spring be released.
Confusing force with energy is a common thread to some geocentrist arguments. "Scientists say the sun keeps the Earth in orbit around it by exerting a gravitational force continuously -- but when I light a stick of dynamite the force of the explosion is expended and then stops. The sun can't exert a force against the Earth indefinitely, it would run out!" Joseph Newman also made this mistake in at least one place in his self-published Energy Machine screed.