Why is mass a form of energy? Why is mass a very concentrated form of energy? Does it have to do something with photons, phonons or nucleus?
 A: The fact that mass and energy are two realisations of the same thing is a direct consequence of the theory of special relativity. Nothing more is required to find this other than Einstein postulates.
The fact that mass and energy are the same thing, although was considered a very strange result at the time, has been countlessly proven experimentally. In accelerators physicists do it all the time: initial particles are given a lot of energy, $17$ TeV in the center of mass frame at CERN for example, smashed together to create countless other particles whose masses can be many times higher than the ones of the particles we started from. 
In a general sense it has nothing to do with specific particles you mentioned, but more with the way our universe is built.
A: Mass is not a "concentrated" form of energy. I'll come to this later but first let's understand the relation between mass and energy. 
In relativity, energy and momentum are parts of a unified object called the four momentum vector $(E, \mathbf{p}$). Now, just like usual vectors in Euclidean geometry, there is a magnitude attached to this vector which is given by $\sqrt{E^2-|\mathbf{p}|^2}$, this is what is mass $m$. Just like the length of a displacement vector doesn't change when you rotate your coordinate system in Euclidean geometry, the magnitude of the four momentum vector doesn't change among different inertial frames in relativity. This magnitude, the mass $m$, thus, is a coordinate independent way to characterize the energy-monentum content of a system. Now, if you go to a reference frame where the momentum $\mathbf{p}$ vanishes, you get that the energy $E_0$ in such a frame which we call rest frame is equal to $m$. 
Thus, mass is the energy of a system in its rest frame, i.e., in the reference frame where the momentum of the system is zero. In other words, $E_0 = m$. 
So far I've done everything in natural units where $c=1$. If you restore the factors of $c$, you'll write $E_0 = mc^2$. The factor of $c^2$ gives the illusion that mass is a concentrated form of (rest) energy because it looks like a small amount of mass corresponds to a large value of rest energy. However, this is purely an illusion created out of our traditional choices of units which are different for energy and mass. In the natural system of units, as I wrote, the rest energy and mass are exactly equal. Saying that mass is a concentrated form of (rest) energy would be like saying length of my arm in meters is a concentrated form of length of my arm in nanometers. 
Sociological edit: Why do people like to believe mass/matter is a concentrated form of energy? 
From comments on this and other answers, it seems necessary to address this point. OK, so mass is what I described above. Now, what does it imply about the mass of a system of particles? Very simple, you go to frame of reference where the spatial momentum $\mathbf{p}$ of the system is zero and the energy of the system in this frame (i.e., the rest energy of the system) is what the mass of the system is. If you are trapped in Newtonian intuition, you'd expect this mass to be the sum of the masses of individual particles. This is obviously not true in relativity, just like other Newtonian things. For example, if you have a neutron decaying to a proton, an electron, and an anti-neutrino, the mass of the the system of the proton, the neutron, and the anti-neutrino would be exactly the same as the mass of the neutron (because mass is conserved) but this total mass would not be equal to the naive summation of the individual masses of the three particles. See, a related answer of mine. So, since the mass of the neutron would not be the same as the sum of the masses of the proton, the electron, and the neutrino, people like to say that some mass of the neutron is now converted to kinetic energy of these particles. Since the difference of the sum of the masses of the resulting particles and the mass of the neutron is a "small" number in the units in which we measure mass and the kinetic energy of the resultant particles would be a relatively "big" number in the units in which we measure energy, people like to say that mass is the concentrated form of energy. But particle physicists use natural units where they measure both mass and energy in electron volts. This would make sure that the difference of the sum of the masses of the resulting particles and the mass of the neutron is exactly the same as the kinetic energy of the resulting particles. So in conclusion, all of this confusion arises because people like to talk using Newtonian intuition and they also like to use unnatural units. :) 
A: I can't agree with the affirmation "mass a very concentrated form of energy", I would agree with "matter is a very concentrated form of energy".
Mass is a property of energy. Of every kind of energy, be it trapped inside the atoms and molecules or not.
Where the mass of the matter comes from? Almost all comes from the binding energy between quarks, inside the protons and neutron in atoms' nucleus. A little comes from the bindings between protons and neutrons making up the nucleus, a little from the bindings between electrons and nucleus, maybe a little from the bindings between atoms to form molecules, and a little from electrons and quarks themselves (I may have missed some sources of energy here).
Why is it so? If by "why", you mean how it was discovered, as Davide Morgante said in his answer, it follows directly from Einstein's special relativity. If by "why" you mean in some grand philosophical sense, you may consider this as physical law, as good as any other.
EDIT about the photon's mass: Despite pragmatic physicists saying photon has zero mass, photons distort spacetime, like all other masses, so much that the whole concept of kugelblitz depends on it. Photons also have momentum and inertia, and the working of light sails depends on it. If you put light inside a perfectly mirror walled box, and weight the box, it will be heavier than without light inside. So please understand mass as "the thing that the scale shows", or "the thing that has inertia" or "the thing that bends spacetime", and photons do have it.
