This answer assumes that absorption of light means the conversion of light to a different form of energy.
If you start with an isolated atom or molecule, the a photon of the right frequency can be absorbed by exciting the atom/molecule to a higher energy state. Why this happens is discussed in my answer to How do photons know they can or can't excite electrons in atoms?. The excited state will decay back to the ground state and re-emit a photon. So the light is not lost though it may be scattered into a different direction.
In a dense medium like a liquid or solid the atoms/molecules interact mechanically with their neighbours i.e. vibrations of the atom/molecule can be transmitted to neighbouring atoms/molecules and vice versa. In these conditions it is very likely that the excited atom/molecule will transfer its extra energy into mechanical vibrations before it has a chance to re-emit the energy as light. The end result is that the light is turned into vibrational energy i.e. heat.
In a comment the OP mentions coloured glass as an example of what is meant by absorption. If we take for example green glass, this is frequently coloured using iron (II) salts. Light is absorbed and promotes electrons between the $\text{Fe}^{2+}$ $3d$ orbitals (the $3d$ orbitals are split into different energies by the environment). The excited electrons relax by transferring energy to the surrounding chemical bonds as vibrational excitations, which turns the absorbed light into heat.