How is sound energy converted to heat energy when it is absorbed by some material? What happens at the molecular level? What exactly is the difference at the fundamental level between both forms of energy?
 A: My take as a physicist who doesn't specialize in acoustics:
While a sound wave coming e.g. from hitting a drum is something that can theoretically be transformed back to macroscopic forms of energy (you could use sound waves to bounce a paper ball up against the gravitational force) due to many particles moving in unison, heat energy is chaotic, "random" motion of many particles which can not easily be "harvested" (See entropy).
The process going from (macroscopic) sound waves to chaotic microscopic motion of particles is turbulence in fluids and any non-elastic processes/collisions in solid state matter (sound waves in crystal lattices a re described as so-called phonons which can scatter of irregularities of the lattice and surfaces). I imagine the biggest factor there to be inhomogenities inside the material, so lattice irregularities, surfaces and so on. Basically imagine light waves being reflected by a rough material. A big wavefront will be broken up into many smaller waves going into seemingly random directions.
It is also possible to atoms/molecules go into an excited state when colliding with each other, therefore transforming kinetic energy of the whole atom/molecule into kinetic and possibly potential energy between it's constituents. The atom/molecule might then randomly emit a photon (light) in a random direction to go back to it's ground state.
There are more processes that could potentially be part of this and which processes dominate the absorption will differ between materials. For example a bucket of sand or some sound-absorbing foam will probably be dominated by almost macroscopic effects of sound waves being reflected (refracted) on the rough surfaces, while in a very pure crystal you might find these very microscopic effects to be important.
Please correct me in the comments if I made some mistakes dabbling in superficial knowledge.
A: 
How is sound energy converted to heat energy when it is absorbed by
some material?

Since the main tag of your post is thermodynamics, I will answer from the standpoint of a thermodynamicist.
First of all, sound and heat are technically not "energy forms". Both sound and heat are forms of energy transfer.  What makes them distinct is that sound is the transport of mechanical energy (energy transfer by work) from one place to another in the form of mechanical longitudinal waves, whereas heat is defined as the transfer of thermal energy from one substance to another due solely to temperature difference.
When @Paul G. says "heat energy is chaotic, 'random' motion of many particles" what he is describing is not heat, but internal energy. In this case, the kinetic energy of the molecules and atoms comprising a substance. The other component of internal energy is potential energy associated with intermolecular forces and bonds. Heat can transfer microscopic kinetic energy between substances, but is not the kinetic energy itself. I nice visual of this energy transfer by heat at the microscopic level can be seen on the Hyperphysics website:
http://www.hyperphysics.de/hyperphysics/hbase/thermo/temper2.html#c1
The above being said, energy transfer by sound can result in energy transfer by heat, but it occurs by way of changes in internal energy.
For example, when sound is absorbed by a substance, it can increases the internal molecular kinetic energy of the substance. The increase in molecular kinetic energy can then result in an increase in the temperature of the substance. The increase in temperature of the substance relative to its surroundings can then result in heat transfer from the substance to the surroundings. The heat transfer out of the substance thus lowers the internal energy of the substance.
However, it should be noted that energy transfer by sound is usually relatively small making the increase in temperature and the heat transfer it leads to very small.
Hope this helps.
A: Interesting question as I looked through several textbooks and don't see this question directly addressed.
Pressure waves in the air (sound) can produce pressure waves in a solid (acoustic modes) when they strike the surface of the solid. To first order these waves are harmonic and can be modeled just as any other harmonic wave, i.e. there is some return force on the atom/molecules when they are displaced from their equilibrium positions, and etc.
In the below, I am assuming a crystal lattice.
Let us assume we have a single frequency sound wave.  This should produce a single frequency acoustic wave in the solid.  However, because of two things, this will change:

*

*Anharmonic terms (e.g. cubic and etc) exist in the equations of motion for the lattice, and


*Imperfections exist in the crystal lattice, such as impurity atoms and dislocations (imperfections in the periodicity of the lattice).
Both of these will lead to excitations of lattice waves of frequencies different than the initial one.  Each one of these new excitations will, of course, will encounter the same anharmonic terms and lattice defects and thus also produce even more and different excitations.
Thus we see now that the original simple harmonic motion in the solid is transformed to many, more random, motions of the lattice - what we would call heat (or, to be more precise with my words, thermal energy, as @Bob D correctly points out in his answer).  Exactly what we should expect.
Energy is of course conserved as the original solid acoustic wave is giving up some energy to these other modes as they are excited - that is, it is damped as it travels through the solid.
All of this could be discussed in terms of phonons.  The square of the displacement of the atoms/molecules from their equilibrium positions is related to the energy of the wave, and can also be shown to be proportional to the number of phonons in the mode (Kittel has a straightforward derivation of this).  Both of the points above are concerned with the scattering of phonons.  Thus, as you scatter phonons out of the original wave and excite other phonons, the original wave's amplitude decreases - it is damped.
A: How exactly sound energy is converted to heat when absorbed by some material is somewhat unclear. It is relatively uncharted by published research.
I have my own ideas about it, but that isn't established science, so I will to the best of my abilities stick to what I know to be common knowledge and call questionable what is questionable. What I will give you are some relevant clues that I do know to be established science. Being careful in that respect, makes this post a bit of a book. My apologies for that.
For starters, not every pressure wave in a gas qualifies as sound. One of the characteristics of sound being its speed. Shock waves may travel much faster than that, since their speed is determined by their source, whereas the speed of sound is entirely determined by the substance it propagates through. Neither the frequency, nor the amplitude can make sound go any faster.
Sound is a substance effect. No substance, no sound. Contrary to light, sound can not propagate through a vacuum. It involves the actual physical displacement of entire molecules. If the speed at which these molecules travel was random, that would not result in sound. In order for sound to propagate by means of displacement of molecules, they all have to move at that same speed. If they would be traveling slower, they would not be able to make the sound wave propagate at the speed at which it does. If they would be moving faster, this would result in dilution of the wave.
So molecules have to already move at that speed, even if there is no sound. Energy wise sound is relatively 'cheap'. It doesn't take much energy to make a lot of sound. However, to increase the speed of a molecule to the speed of sound and back again to a relatively low speed with every passing wave, would involve way more energy considering the total mass that is being displaced. One could conclude from that, that the speed of sound is the internal speed of the gas it propagates through. That however as much as it may make sense, is not established science.
So what determines the speed of sound? The answer is fairy simple, its the energy that causes it. So it must be the consequence of an inter-molecular effect that releases that exact amount of energy. This whole kind of energy release is caused by something called a trigger effect. A trigger effect occurs when the balance between two (or more) opposing forces reaches a critical point at which they suddenly swap dominance.
Allow me to give you an example to illustrate the principle of how the energy causing constant speed is kept constant and how they directly relate to each other..
Imagine a magnet in the form of a pipe. To one side of the pipe, we hook up an air compressor. Then we take a flat piece of steel plate and try to attach it to the open end of the magnet. That is the end where the compressed air comes out. As long as the pressure coming out of the compressor is high enough, it will be impossible to attach the plate to the pipe so that it stays there. Now we reduce the pressure until it becomes just possible to make the steel plate stick to the open end of the pipe magnet, sealing it of, stopping the airflow. Now we can pull of the steel plate and push it back on again as often as we like. Notice how the airflow pushes the plate away from the pipe, making it hard to push it on, but as soon as the magnet 'grabs' it, you don't have to push any more.
The power with which we push the plate on to the pipe may vary. We can gently push more and more until it holds, but we can also just slam it on with a lot of force. However, if we measure it, we will find that the power needed to pull it off is always the same. As long as that force is not applied, either by raising the pressure or pulling the plate, the plate will stay on the magnet, regardless of how long it is left there. But as soon as that exact force is applied, it will always again immediately come of. It is a bit harder to measure, but if we can, we will find that the plate when pulled off, separates at the same speed every time too. This is how constant speed is created by use of air pressure and an opposing magnetic force. A repelling and an attracting force over one dimension.
Are there such forces in play between molecules? Oh, absolutely. Gas is pretty much always somewhat pressurized, which constitutes a repelling force and mass attracts mass, which is an attracting force, just to name a few, but there are more forces active on an inter-molecular scale.
This is where it gets tricky. You see, gas molecules repel each other, the way our force pushing the plate onto the pipe was repelled by the air pressure coming out of it. However, it only leads to a constant separation speed when the trigger function is applied. Between molecules that would mean they would have to actually touch each other first, like in a liquid. There seems to be little research done on that, making the whole idea controversial to say the least. So far I have neither been able to get that idea confirmed, nor refuted by argument. Rejected yes, ignored, discarded, ridiculed, you name it, but not refuted.
It would answer your question. If sound would actually invoke condensation followed by evaporation resulting in a constant speed, than that would include heat transfer. Heat transfer by conduction in liquids occurs, even if it is just between two molecules. What we do know is that pushing air beyond the speed of sound causes condensation. That is clearly visible as jet fighters go into trans-sonic flight. But that doesn't prove anything.
