Energy and its physical significance 
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*What does it exactly mean when we say that energy has moved from one body to another, what has physically been transferred? The concept of energy is very confusing, please help.

*And if gravitational potential energy is stored in a body, does the body's mass increase according to $E=mc^2$?
 A: What is energy?
I'm going to start by using a very basic understanding of energy (one suitable for a first year course):

Energy is any property of an object or system that allows it to do work.

Here "work" is used in the sense of force applied over a distance: $\mathrm{d}W = \vec{F} \cdot \mathrm{d}\vec{s}$.
What kinds of properties would qualify this way?
Well, a moving object could (it things were arranged right) ram into a nail and drive it into a wall. As that requires force applied over a distance work is done and the moving object loses speed as it drive the nail---the energy of motion (kinetic energy) gets turned into work, and the slower object now has less ability to do work.
A heavy mass at the top of a funicular can lift what every is placed on the other side, work done by virtue of position in a gravitational field. Gravitational potential energy is traded from one object to another. Note that the counter weight of the funicular doesn't "contain" the energy on it's own, rather the combination of the weight and the planet has the ability to do work (if you take the funicular into deep space it won't lift anymore, because the Earth was an necessary part of the system) so it is the system that "has" the energy.
A stretched or compressed spring contains energy (elastic potential energy), as does a confined sample of hot gas (thermal energy), or a gallon of kerosene (chemical energy), or a ton of uranium ore (nuclear energy).
Caveats
Some of the other posters have offered more sophisticated definitions of energy, but they agree with the one I have given here when put into a simple mechanical context. On the other hand you can't get to a Noetherian conception of energy from the definition I have given here without a leap of intuition.
Also thermodynamics tells us that energy can exist in forms where it's "ability to do work" is effectively neutralized by existing in a high entropy arrangement.
Energy and Mass

"if gravitational potential energy is stored in a body, does the body's mass increase according to $E=mc^2$?"

Again, it's not in a singular object but in the arrangement of objects in a system, but yes if you act on the system as a whole it's mass is different from the sum of the masses of the parts by the potential energy. (But there is a twist here, for gravitation that energy difference is negative because you test the mass of each of the parts at infinite remove.)
But do compute the energy difference between you and the planet at infinite remove and and with you on the surface, then convert that to a mass difference. It is surprisingly small. On the other hand, nuclear fission and fusion convert measurable fraction of the starting mass to forms of energy we can (in principle) use, and the very protons and neutrons of your atoms are more binding energy than mass of their parts.
A: Energy is a usefull quantity that can be measured.It is conserved in a closed system to.
There are equations of motion which are given in terms of energy, called Euler-Lagrange equations. It is simply an observation that interacting systems effect each other in such a way as to transfer energy from one to another, and we can observe that exact amount lost in one system equals amount gained in other. Two basic forms of energy are kinetic and potential and every other energy, like electric or thermal, are basicly these two basic ones.So, it is the energy in motion, kinetic, and the energy of a body at rest in some potential/force field. It is simply a kinetic energy that the body will certainly aquire if we let it move, like water falling down the clif. Formula for kinetic energy is:
E=0.5mxv^2
This quantity has all of these nice properties I just described. So to really know the energy, you have to observe it in its natural environment, equations of motion in physics. Then it gets clearer and clearer. 
A: For most everyday purposes, E = mc² is largely irrelevant.
Consider 1 litre of water at 0°C.  This weighs approximately 1Kg.
If we heat it to 100°C, then we have added 418600J of energy (the specific heat capacity of water is 4186J/Kg°C).  Applying m = E/c², we find that we have made the water 418600/(300000000²) = 0.0000000000047Kg heavier.
A: Energy is not a physical property. It is an abstract mathematical quantity that turns out to be very useful to calculate because it tells you important information about the state of the system. There are many kinds of energies associated with motion (kinetic energy), position (potential energy), mass ($E=mc^2$), and others. When physicists speak of transferring energy from one system to another, it is shorthand for talking about some physical processes changing the state of two systems. In many interacting systems, Newton's Third Law states that interactions have opposite effects on the two systems: one speeds up, the other slows down. The overall effect is to keep the total energy associated with motion constant. In fact, physicist often choose a definition for energy so that it stays constant to make calculations easier. Different kinds of energy can be converted into others as well.
As an analogy, think of profit in business. I buy a car for \$5,000 and sell it for \$7,000. My profit is \$2,000. But, where was the amount of \$2,000 ever involved in the transactions? It wasn't. Profit is an abstract quantity of money that is useful to calculate since it tells you important information about the state of your business.
As for mass changing with energy, this is a subtle question. A moving object does not gain mass from kinetic energy. Heating an object does. You can think of mass as the total energy of a system as measured by an observer that is not moving with respect to that object. If the object is moving, the energy due to motion has to be subtracted away to get the mass.
A: energy can be seen as something that may change the state of a system. you can use energy to move a stone from point A to point B. to do that you need to apply a force (some forces can be related to the gradient of a potential). in the case of the gravitational potential, the related force is the gravitational force. therefore the gravitational potential energy is the energy that can change the state of a stone. if it's a a point A, say at a hight $h$, it will have a potential energy $mgh$. This energy may be "released" to change the state of the stone from being static to being in a free fall state. 
when we say that energy is transfered or exchanged, we mean that the energy doesn't disappear. in the case of the free falling stone, the potential energy is transformed into a kinetic energy. the stone still has the same energy but its "nature" has changed (the kinetic energy isn't related to the gradient of some force). 
when the stone hits the ground, the energy is transferred to the surrounding (meaning the earth) by making some noise, heating or deforming the impact area... if the stone falls on a trampoline, the kinetic energy of the stone goes into the potential energy of the trampoline (by elastically deforming it).
this is not to be confused with $E=mc^2$. very roughly, what this formula says is that mass is energy.
A: 
I) What does it exactly mean when we say that energy has moved from
  one body to another, what has physically been transferred?

Contrary to the view expressed above by Mark H < https://physics.stackexchange.com/a/217505/76263 >, I think energy is a fundamental physical property and not just “an abstract [useful] mathematical quantity”.
In fact, the energy of a system and how it varies, yields the physical behaviour of the system in terms of a set of states. Of course, physicists make the concept of energy and its change quantitative by an abstract mathematical formulation. Usually, having a quantitative form of the energy plus some evolution equation (‘equation of motion’) will provide a complete physical description of a system in terms of a set of physical states which are appropriate for the system in consideration.
The “transfer of energy” from a body/system to another is the most fundamental way we can describe a physical process. This leads to the change of the state of the body/system. As an example, the process of heating of water can be described as transfer of “thermal energy” from the heater to the water.
You can find a very nice intuitive picture in “The Feynman Lectures on Physics”, volume I, section 4-1. I cannot do better.

II) If gravitational potential energy is stored in a body, does the
  body's mass increase according to E=mc2?

This question is rather vague, if not ill-posed.
First, let's assume that we confine ourselves to the special relativity  (where the famous formula, $E = m c^2$, comes from), and treat the gravitational energy as any other energy – the gravitational force is in fact very subtle.
Having this in mind, one has to distinguish properly between two types of mass (at least):
Firstly, a massive body has a ‘rest mass’ $m_0$ defined as the body's mass as seen by a stationary observer in the reference frame of the body. This is an invariant property of the object.
Secondly, there is a ‘relativistic mass’ defined as
$$ m_{relativistic} = \frac{m_{rest}}{\sqrt{1 - v^2/c^2}} $$
that is higher compared to the rest mass, from the viewpoint of an stationary observer which is moving relative to the frame of reference of the body. This concept of ‘relativistic mass’ is rather obsolete, since usually one prefers to ascribe an invariant mass to a physical body.
For a more detailed explanation, see this post: < What's the difference between the five masses: inertial mass, gravitational mass, rest mass, invariant mass and relativistic mass? >.
Within the limits of the discussion above (special relativity + Newtonian gravity), the gravitational energy of a body would be a contribution to the rest mass only.
