Calling light either a particle or wave is incorrect. It's neither. However, modeling light mathematically as either a particle or a wave is very useful for many things. Never confuse a model for the thing itself.
Like waves in water, or sound waves, light will demonstrate the phenomenon of interference and diffraction.
One of the first people to posit a wave theory was Christiaan Huygens:
Christiaan Huygens - Wikipedia, the free encyclopedia
How did he decide light must be a wave? Careful observation of obscure things. Like double refraction and observation of the 'airy disk' seen through a telescope
See some of the images of Huygens principle:
Huygens
Issac Newton had a competing 'Corpuscular', theory of light:
Corpuscular theory of light - Wikipedia, the free encyclopedia
Newton's theory is still with us, except 'corpuscular' is now 'photon'.
An example of how weird this all gets is the famous 'double slit' diffraction experiment. You can easily duplicate this with a laser yourself. I really recommend you watch this five minute video to see it in action (> 1.5 million views):
YouTube - Dr Quantum - Double Slit Experiment
Is it a particle or a wave?
For example, in optics, diffraction limits how much you can magnify something. A reflecting telescope has a primary mirror. In order to produce a good image the 'figure' of the mirror should not deviate by more than 1/8 wave from an ideal parabola. 1/8 of what you might ask? 1/8 wave of whatever wavelength of radiation is being observed - usually green/yellow (sodium) light. A pretty small tolerance of error. When the figure is good the telescope is said to be 'diffraction limited', meaning there's no need to polish and figure the optics beyond this point. However, even when the mirror has an ideal figure diffraction means than an in focus star looks something like this:
It's not a perfect disk, rather it's more of a splash, with concentric rings around a bright central disk. When you look at a star, either naked eye or through a telescope, you're not seeing the surface of the star - it's much too far away. Rather you're seeing an optical phenomenon - the 'Airy-disk':
Airy disk - Wikipedia, the free encyclopedia
The larger the telescope's mirror, the smaller the airy-disk becomes, and its resolving power increases.
Another example of a device that uses the wave nature of light to do useful things is an 'interferometer':
Michelson interferometer:
Michelson interferometer - Wikipedia, the free encyclopedia
Laser speckle is yet another example of interference. Shine a laser into a corner of the ceiling and see the 'speckle' or 'snow' all around the dot. You can't focus on it, it's always sharp. The space of the room is filled with an interference pattern.
Finally, here are some images I took shining a red laser off a frosted incandescent bulb. The laser was a few meters away from the bulb, and projected spot a few meters away from the wall. The images are an interference pattern created by the surface of the lightbulb, and represent a sort of map of it's roughness. Interference. A laser magnifying glass.
http://www.fractalfreak.com/wx/interfere1.jpg
http://www.fractalfreak.com/wx/interfere3.jpg
http://www.fractalfreak.com/wx/interfere5.jpg