This experiment has two important outcomes, both of which are highly confusing, and demonstrate that the universe does not work in ways that we are equipped to understand.
To set up this experiment, you get a light, a photographic plate, and a piece of cardboard with two slits in it. You set up the cardboard so that the only way that any light can reach the photographic plate if it goes through the slits.
You then shine the light though the slits. Instead of getting neat little slits of light on the photographic plate, you get an interference pattern. This shows that the photons of light are waves, and that the waves are interfering with each other. Some waves strengthen (amplify) each other, giving you bright bars, and some waves weaken each other, giving dark bars. This is elementary physics; nothing mysterious about it at all.
The interesting bit is, you can see the pinpoints where each photon hit the photographic plate. The interference patterns are made up of hundreds of individual photons impacting the screen. When they hit the screen, they don't act as waves, they act as particles. Photons travel as if they are waves and arrive as if they are particles.
It gets more confusing. Redo the experiment, but this time you let the light out one photon at a time. You will still get an interference pattern. There is only one photon, but it still acts as if other photons' waves are interfering with its wave.
One hypothesis (I believe it is the strongest), is that the photon is interfering with itself. Quantum mechanics says that particles can exist in multiple 'life courses' until they are observed, and then they collapse into one course when observed. As you do not observe which slit the particle goes through, it literally goes through both. The photon is interfering with it's own other possible existences. Yes, it's weird. But it appears to be true, and it seems to be representative of how all small particles work.
For a more technical description, see Young's Slits