Why do particles vibrate? It seems like all molecules vibrate and they vibrate more when heat is added. Why do they vibrate in the first place? And why does adding heat to a system make its molecules vibrate more? What are the forces involved in making particles vibrate?
 A: All molecules do not necessarily vibrate. It depends on the energy that is absorbed by the molecule.
In the case of absorbing electromagnetic energy, each portion of the electromagnetic spectrum has quantum energies appropriate for the excitation mode of the molecules. If there are no available quantized energy levels which match the quantum energy of the incident radiation, then the material will be transparent to that radiation, and it will pass through.
The quantum energy of microwave radiation, for example, corresponds to the excitation modes of torsion and rotation of polar molecules such as water.  It is too low to cause vibration.
The higher quantum energy of infrared radiation, on the other hand, corresponds to the excitation mode of molecular vibration.
Moving up in energy to visible light, the quantum energy of visible light corresponds to changes in electron levels. .
For a more detailed description of the interaction of radiation with matter, check out http://hyperphysics.phy-astr.gsu.edu/hbase/mod3.html

So if I heat water in a microwave, the water get hotter because the
water molecules are "torsioning" and rotating, but not vibrating?

Water will get hotter (increase in temperature) due to either rotation or vibration. But for microwave radiation it is due to rotational KE. When those rotating molecules collide with others that are in translation, the rotational kinetic energy is passed to them converting to and increasing their translational KE and thus increases temperature. Keep in mind it is translational KE that determines temperature. As stated in the website I gave you, the rotational kinetic energy is, in effect, randomized.
The same situation occurs when the molecules vibrate due to absorbing infrared radiation giving them the higher level vibrational kinetic energy. Collisions between those molecules converts vibrational KE to translational KE increasing temperature.
The main difference between microwave radiation and infrared radiation is infrared radiation is more strongly absorbed by the molecules. This results in infrared radiation potentially increasing the temperature of food more than microwave radiation.
For example, if you broil food in a conventional oven the infrared radiation can raise the temperature of the surface of the food enough to burn (char) it. That generally doesn't happen in a microwave oven. Microwave ovens essentially boil food. That's why some microwave ranges use broiling elements (infrared radiation) in addition to microwaves in order to get a desired browning effect.
Hope this helps.
A: The particles in a solid vibrate because the vibration of surrounding particles causes fluctions in the force fields which maintain the structure of the solid.  The vibrational energy may be in thermal equilibrium with the environment as particles at the surface may be subject to collisions from a surounding liquid or gas.
