The photoelectric effect clearly demonstrates that light can behave as we expect of particles, but diffraction experiments demonstrate that light can behave as we expect of waves.
Electrons though are little bits of matter, and they too display both wave and particle properties. A beam of electrons pointed at two slits will display inteference effects between the two beams.
More surprising is that if electrons are shot though the slits one at a time, an identical diffraction pattern is built up. It seems as though the electron is interfering with itself.
The relationship that describes wave particle duality is the De Broglie relation
All particles have wave properties, and light and similar waves display particle properties described by the De Broglie relation. In fact both 'light' and 'particles' are little 'wave packets' occupying a certain region of space as shown below.
