Origin of Elasticity Why is it that not all bodies possess Elastic behavior? What is the origin of elasticity or plasticity? I mean, it's a physical property. So, how does it relate to atoms or molecules in different phases? (It should have some relation with atoms)
 A: A toy model of a solid
A very simple model of a solid is to imagine a bunch of molecules linked to their nearby neighbors by springs (you don't need to imagine a crystalline lattice, it can be amorphous).
The springs are the effective electromagnetic interactions between the molecules; they are strongly repulsive at close range and attractive at modest ranges which implies a zone of equilibrium. And for small displacements they really are roughly linear which means that the spring model makes sense.
The model is not perfect of course, at overly large distances the interaction drops strongly and the springs effectively break; and relative motions can cause  new "nearby" neighbors from behind old ones rearranging the tangle of springs entirely.
Given time the solid will settle into a global equilibrium disturbed by the thermal motion of all molecule; we assume a temperature low enough that each one jitters around in it's own little space trapped by the forces from it's neighbors.
Elasticity
Now, if you push slowly on opposite sides of this thing, each surface molecule will move a little towards their inner neighbors who will move a little out of the way and the force will becomes distributed throughout the bulk with the distance between molecules reduced slightly in the direction of the compression. Pulling the two ends apart spreads the molecules slightly.
In either case, if you relax the external forces the body returns to it's original configuration.
Ta Da! Elastic behavior!
How about in-elasticity?
If molecules get pushed too hard they might pop over a local maximum and settle into a new, different equilibrium configuration. These kinds of large displacements can result in the breaking of some springs and the establishment of new ones. 
If we release the pressure on this configuration it will return to a different free configuration than our starting point: non-elastic behavior.
This model has too much hand waving in it to produce a clear taxonomy of non-elastic behaviors.
Fluids
Because the molecules of liquids and gases do not maintain their relative positions this model is completely inappropriate and not applicable to these phase of matter.
A: Elasticity, malleability, fragility, tensile strength, compressive strength, and the phases of matter are all determined by intermolecular forces.
Once a substance’s constituent particles lose sufficient kinetic energy (thermal energy), the intermolecular forces will be able to overcome the elastic transfer of momentum during a collision between to molecules.  Those intermolecular bonds can be rebroken with a sufficient disparity of kinetic energy applied the the constituents of that molecule.  Like gravity holding a golf ball to a tee, kinetic energy transfered to the ball via a club will separate the ball from the tee.
Now imagine a different regular shaped golf balls:  tetrahedral, cubic, ...  the more sides that are added the closer to a sphere it becomes.  The fewer sides, the more likely they will settle into a stable lattice.  This illustrates one aspect of variability in intermolecular forces.
Another aspect would be akin to varying the size or weight of each polyhedron.  With shape, size, and weight, you can adequately describe many material properties.  Placing them in a vertical wind tunnel, you can simulate increasing thermal energy.
Low airflow will be insufficient to jar any shaped polyhedron enough to shift its position relative to the others.  As airflow increases, polyhedrons closer in shape to spheres will begin to shift.  These are more MALLEABLE than the lower n-Polys.  The lower n-Polys are more elastic because they are able to shift enough to transfer the kinetic energy without shifting far enough to change locations.
Now, at the bottom of the chamber there is an array of equidistant golf tees that rise and fall.  They all move in unison, and the amount they move is directly proportional to their distance from the center of the chamber.  This simulated tension.  The height they rise portrays the variation in tension along the the axis perpendicular to the vector of the force applied.  High n-Polys are able to maintain contact despite the variation in tee heights, and they will shift sooner than the more regularly aligned low n-Polys.  On the other hand, low n-Polys will maintain structural cohesion until a threshold is reached that causes the structure to violently reassemble all at once.  That demonstrates high tensile strength.  If the tees recede and the structure remains infancy with no change in location for any Poly, that demonstrates elasticity.
Now, leaving the tees just below the threshold of collapse (break point), but turning up the wind (thermal energy) will demonstrate how thermal energy will cause random Polys to shift from their local position.  As each Poly settles into a new equilibrium, the threshold for them to shift rises once again.  This random shifting demonstrates a spring under tension shifting its equilibrium away from the original shape and toward a new shape.  The Polys atop the central tees being the closest to the threshold for shifting positions will shift away from the center relocating to a lower tension equilibrium position.
And for fun, turning the wind up high enough causes the polys to first shift locations freely within the tunnel (liquefaction), then, with enough wind, cause Polys to bounce around in the tunnel; off the walls and each other.  Lower in the tunnel, Polys are closer together, and up higher they are farther apart.  That demonstrates density gradients in an atomosphere.  With enough wind, they will start ejecting fromnthe tunnel, and that demonstrates molecules of gas attaining escape velocity causing them to leave the atmosphere.
