The most famous experiment demonstrating the wave nature of light is the Double Slit Experiment. Its genius lies in its simplicity.
Thomas Young set up this simple experiment back in 1801.
Young shone a lamp emitting only one wavelength (color) through a screen with two narrow slits cut into it. Behind the first screen Young placed a second, viewing screen to see how light came out after passing through the two slits. This second screen kind of "catches" the light that passed through the two slits. He looked at the result and was amazed. What sort of pattern does light make on the second screen after passing through the slits?
If light were (only) a particle, we should just see two bright lines that match the first screen's slits on the viewing screen. We can imagine it as taking a spray paint can and spraying paint through the two slits. Most of the paint would stick to the outside of the slit screen, and some of the paint would make it through the slits and hit the second screen.
Well, that's not what Young saw.
What did he see instead? He saw alternating lines of light: light, dark, light, dark…and so on. But particles don't interact that way, particles behave like the paint would, making only two lines. Particles do not spread out from the slits the way that Young observed, either, but waves…waves do that.
The reason we see the light-dark-light pattern is because of areas of constructive and destructive interferences. Also, waves, unlike particles, diffract. When the light waves squeeze themselves through the slits on the screen, they diffract around the edges of that slit. The slits essentially create two circular wave fronts, free to interfere with each other and create those bands of light and dark.
Check out https://commons.wikimedia.org/wiki/File:Young_experiment.gif of a Double Slit Experiment. Wherever waves interact, or interfere, constructively, the wave becomes dark orange in this movie. Wherever the waves interfere destructively, and cancel each other out, you get yellow. When we look at the wall on the very right of the movie, we see spots of dark orange hitting the wall: these correspond to spots of light. We also see spots of yellow on the wall and these are the dark lines resulting from waves that cancel each other out.
When two waves meet they interact with each other. As long as they're the same sort of wave they just add themselves together. If two waves meet peak to peak, we get a wave with a peak twice as high (twice the amplitude). If they meet peak to trough, we get a smaller wave, or no wave at all if the two waves have the same amplitude.
To fully understand this concept, please try watching the video below: