I'm trying to understand the process that converts light into heat. I found poor explanation that do not include the whole process.
How any atom, or most likely any molecule, reacts when a photon interacts with it, depends on the structure of the molecule. The incoming photon has a certain energy and, depending on this energy, it may cause the molecule to vibrate, producing heat.
If you look at the diagrams above of a two atom molecule, the springs represent the bonds and, if the incoming photon has the right amount of energy, it may cause vibration or rotation of the molecule, producing heat.
It's all down to conservation of energy and momentum. The relatively high energy incoming photon, say an infrared frequency photon, interacts with the molecule. The energy available from the incoming photon is split between moving the atoms apart (which takes energy). The moving atoms may then produce a photon in a lower frequency infra red part of the electromagnetic spectrum, which you feel as heat.
Are the vibration caused by the change in charge between the molecules?
I assume that the atoms are not ionized, since it would cause the matter to conduct electricity, wouldn't it?
The system is based on simple harmonic motion, the atoms use kinetic energy to move closer together, and then mutual repulsion pushes them apart, like a mass on a spring.
The atoms can be ionized, but that takes higher energy photons to remove electrons, rather than this case, where infrared radiation is enough to cause vibrations, but not ionization.
I write the following few lines because I feel you want to understand this problem in as much detail as you can. The links are to articles on Wikipedia.
There are three different levels at which we can understand most physical problems, in order of accuracy of prediction and intuitive understanding, these are:
Classical mechanics, which you have obviously gone beyond in this case.
The second is Quantum Mechanics, which is what your question is based on, although you are still using classical concepts, so when you say "a photon hits an atom of a molecule", that is not how Q.M. describes things.
The third, which models interactions such as the above to give us the best picture of "what really happens", is Quantum Field Theory.