This is a profound question and I will not delve too deeply into the issues but rather, will simply point out some interesting aspects of light and some questions about light.

I noted that light (EM radiation) has wave-like properties. However, there are some experiments where light has particle-like properties. The core of the problems with our understanding of light concerns this duality in its nature.

This is a conceptual problem because waves are spread out in space, while particles are like BB's, that is, concentrated to points in space. There are some nice experiments which bring these points home.

• Diffraction and Interference

  

  The interesting thing is that light behaves like a wave when it shines upon a single slit or a double slit. That is, it doesn't seem to act like a stream of particles.

• Black Body Radiation
  
  Planck was only able to understand the results of radiation experiments by making the radical assumption that the light came in discrete bundles of energy of size
  Energy = Planck's constant x frequency = h x c / W

  where h is Planck's constant. Note that f x W = c and the photon's energy is large for high frequency or short wavelength radiation. The packets of light (energy) are referred to as

  photons

  A water wave or sound wave can have any energy depending upon its amplitude and so this notion is odd for a wave.

• Photo-electric Effect
  This is an another example of an experiment where light behaves as a particle, not as a spread-out wave.

  

  In this experiment, EM radiation is directed at the metal plate on the left. For low energy light (infrared radiation), nothing happens. Even if the intensity of the beam is increased (increasing the number of photons), nothing happens. However, if a beam of higher energy light (ultraviolet radiation) is shined on the plate, a current flow develops. The UV radiation apparently ejects electrons from the atoms in the metal plate. Even for a very low intensity of UV light, a current flows. Why?

• Double Slit Experiment
  The results of shining high intensity beams of light on slits are interesting. When a low intensity beam of radiation strikes the screen, it interacts one at a time, i.e., it interacts like it is made of particles. However, if we wait long enough, the pattern which gets built up on the screen is the one predicted by the wave model, i.e., the light behaves like a wave.

Light cannot be both a wave and a particle. What apparently is happening is that light travels through space (propogates through the Universe) as if it were a wave, but when it comes time for the light to interact with matter (exchange energy and momentum with the matter), it behaves as if it were a particle in that it exchanges energy and momentum in discrete chunks. This is telling us that light is neither a wave nor a particle (it is something else).