(Sorry rewebster and genneth, but I'm going to use "is" in this, because the easiest way to describe the photon model of light is to say "a photon is").
The photon is a single particle in the particle model for light.
Before quantum theory, and the concept of wave-particle duality, the nature of light was disputed. Some argued in favor of the particle model, others argued in favor of the light model.
Young's double slit experiment seemed to verify that light was a wave, until it was discovered that the same effects could be seen if light was let out very weakly (one photon at a time). The double slit experiment showed that light behaves occasionally as a particle (photon), and occasionally as a wave.
An electromagnetic wave consists of an electric field and a magnetic field perpendicular to one another. Moving electric charges create magnetism, while moving magnetic fields induce electric charges. They can travel across vacuum because it is self-propagating in this way (each field produces the other). It doesn't require a medium, because it is its own medium (hope that's accurate).
I don't know what your source means by "displacing matter." Photons have no rest mass (they do have mass proportional to their wave energy while moving).