What fundamental physical property determines whether something is transparent or not?

Question

What is the most fundamental difference between, let's just say, glass and wood? Why can light travel through glass but not through wood?

Answer 1

Whether it is an electrical conductor at optical frequencies.  If it is an insulator, it is transparent. If a conductor, the E-field is shorted out, and the light is reflected.  If a semiconductor, the light can propagate a short distance into the material, but its intensity will decay exponentially.