What is energy?

"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 of 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 anything 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."

Energy conservation is a fundamental principle of physics.  Energy is always conserved.  It may change forms, but the total amount of energy in an isolated system is constant.  Energy can, however, be converted from one form to another form.  There are many different forms of energy.  The two major categories are kinetic energy and potential energy.

Kinetic energy is the energy of moving objects. 
Examples are:

• the ordered kinetic energy of moving macroscopic objects
• disordered kinetic energy or thermal energy
• electromagnetic energy and radiant energy

Potential Energy is stored energy which can be converted into kinetic energy.
Examples are:

• Gravitational potential energy
• Electrostatic potential energy
• Stored Mechanical energy (springs, rubber bands, etc)
• Chemical energy (microscopic potential energy)
• Mass energy (nuclear energy)

The SI unit for energy is kg m²/s² = Joule (J).

An object of mass m moving with speed v has kinetic energy.  We define the (translational) kinetic energy of the object as K = (1/2)mv².

Kinetic energy increases with the square of the speed.  When the speed of a car is doubled, its kinetic energy increases by a factor of four.

The gravitational potential energy U_g of an object with mass m  near the surface of the earth increases linearly with height.  We write U_g = mgy, where y is the distance above a reference point and g is 9.8 m/s².  The zero of the gravitational potential energy, i.e. the reference position, is chosen arbitrarily.  However a difference in gravitational potential energy

ΔU_g = mgΔy = mg(y_f - y_i)

is uniquely defined.

The gravitational potential energy can be converted back into other forms.   If we let the object fall towards the ground it will loose potential energy and gain kinetic energy.

The potential energy U_s stored in a stretched or compressed spring with spring constant k is elastic potential energy.  U_s = (1/2)kx², where x is the displacement from equilibrium, if the reference point is chosen to be x = 0.  (U_s only depends on the position of an object, not on how the object reached that position.)

Matter itself is a form of energy.  The amount of energy stored in a piece of matter at rest is given by one of most famous formulas in physics.

E = mc²

The energy intrinsically stored in a piece of matter at rest equals its mass times the speed of light squared.  A very large amount of energy stored in very small amounts of matter.  Only a small fraction of it can be converted into other forms of energy in our universe.

Microscopically all energy is either kinetic or potential energy, and, if the system under consideration is the whole universe, the total energy of the universe is conserved.

Work is the conversion of one form of energy into another.