What determines whether or not photons can be absorbed by matter? Intuitively, the answer is that a charged particle can absorb photons (whether it's positively or negatively charged), and neutral particles can't. But light is often absorbed by atoms in bulk matter, increasing their thermal motion. The problem with this is that atoms are neutral, and so you wouldn't think that they could absorb light.
It is the other way around.
An isolated, point-like charged particle like an electron cannot absorb a photon. It is forbidden, because energy and momentum cannot be simultaneously conserved. Only scattering is permitted in such circumstances.
However, a charged particle that is within the electric field of another (e.g. an electron in the electric field of a nucleus) can absorb (or emit) a photon.
The way to think about this classically is in terms of electric dipole moments. An atom has no net charge but is can have an electric dipole moment, because the positive and negative charge are not in the same place. This electric dipole moment can interact with incoming electromagnetic waves and can be made to oscillate by extracting energy from the incoming wave.
In the quantum mechanical extension to this picture, the electric dipole will only oscillate in certain modes with discrete energies. Only photons within a certain narrow range of frequencies/energies will be able to excite (or de-excite) these oscillations.
The extension to bulk matter (and by that I assume you mean solids) is that the atoms themselves are arranged into lattices and other structures. These have electric dipole (or higher order multipole) moments that can be excited into oscillation by electromagnetic waves/photons in a similar way. i.e. They offer further vibrational modes that act as a pathway to the absorption and emission of photons.
Nuclei themselves can have electric dipole or higher order multipole moments and are able to directly absorb (or emit) gamma rays.
It's the electrons of an atom that absorb the energy of the photon, not the nucleus. The frequency of the incoming light wave is at or near the energy levels of the electrons in the matter. The electrons will absorb the energy of the light wave and change their energy state.
Atoms are perpetually vibrating at any temperature above zero Kelvin. Some of those atoms vibrate sufficiently vigorously that their vibrational energy is roughly equal to the electric energy absorbed from the photons (essentially, they are in resonance with the photon energy). Those atoms then make a quantum transition from electronically excited to vibrationally excited, meaning that the energy causes the whole atom to move. We feel that motion as "heat." The atoms which make the jump to vibrational excitation soon collide into neighboring atoms, dissipating their vibrational energy.
As @Codosaur mentioned earlier, it is the electrons that absorb the the energy of the photon and not the Nucleus.
Let's take a theoretical example to see how this works.
Imagine an atom without any disturbances. It's electrons are in a relaxed state. That is, the electrons are stable in their orbits.
Now, a photon's energy is absorbed by the atom. This can possibly lead one (or more) electrons of the atom to get excited and go to a higher energy level.
When the electron goes to this excited state, it is unstable.
To those who don't know, when an electron goes from one energy state to other (higher or lower doesn't matter), it is called a transition.
In order for the transition to happen, the energy that is already present in the electron should be greater than or equal to the difference between the 2 energy levels.
The part after this is only if you are interested. It is not directly related to the question.
After becoming unstable, the electron will soon come back to it's normal stabilised state and will also give out a Photon with an energy of the difference between the 2 energy levels of the electron.
Hope this helps,
P.S. This Link has some really nice visualisation of the same.