Please allow me to relate this to something you have likely experienced. In most science classes, at some point in time, they do the whole bit with the projector and the piece of cardboard with the slit cut into it on the right. When the room is darkened, and the projector light shown through the slit, you get a vertical narrow beam of light smoothly spreading out into darkness all about it on the right side of the screen.
Then the teacher typically flips the piece of cardboard upside down so the slit is on the left side. The same thing, of course, happens except the vertical narrow beam of light is now on the left side of the screen.
Then the teacher asks the class what will happen if they put a piece of cardboard up with two slits cut into it, one on the left, and one on the right. Practically everyone who has never seen this demonstration will reply that they expect to see two narrow beams of light one on the left, and one or the right, with maybe a bit of overlap with each other.
However, that is not what happens. What actually happens is that you see a field of alternating bands of light and shadow across the whole screen.
What is happening here is that when there is one slit, the beam of light from the projector acts like a stream of particles, and you get a typical spread pattern looking like a vertical narrow beam of light that smoothly gradates into darkness all around it. But when there are two slits, the beam of light acts like a wave form, and the bands of alternating bright and dark are the wave pattern, from the top down. The bright areas are the peaks of the wave, the darker bands are the troughs.
This is a simple pair of experiments, that anyone can perform - and I have seen for myself, demonstrating the dual nature of light. More complex and precise experiments such as those mentioned by Manna, prove that what I have simplistically described in layman's terms is actually what is going on. Please forgive any loss of fidelity due to simplification.
The really interesting part about the simple experiment described above, I am told; is that if you place a photon counter right up against one of the slits, the beam of light reverts to the particle stream and projects the two bars of light that most people expected in the first place. Which apparently not only proves the dual nature of light, but that atoms are somehow aware of being observed, likewise proving the "observer effect".