The double-slit experiment shows that it acts as both a wave and a particle. Look that up on google or something for more information.
If we analyze a beam of light with a double slit or a diffraction grating, we get an interference pattern (colored light fringes) which is consistent with the wave nature of light. We can perform measurements on the interference pattern and calculate wavelengths. That is relatively easy.
1. Black body radiation:
The particle nature of light was originally used as a hypothesis to explain why a hot glowing object does not give off light of equal intensities at different wavelengths (colors). Classical physics predicted That the energies emitted by light produced by a hot object at different colors should be equal. However, when a hot object just starts glowing most light emitted is infrared and red. The intensities decrease at higher frequencies (green, blue, violet, ultraviolet, etc).
2. Photoelectric effect:
When light shines on a metallic surface, electrons are ejected. "Electric eye" devices are based on this effect. The number of electrons ejected depends more on the wavelength than the intensity of the light beam. Einstein showed mathematically how particles of light (photons) can explain this better than waves.
A couple of links for the particle nature of light that are relatively easy to follow:
(Broken Link Removed)
The first link improved: