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 source, a photographic plate, and a barrier with two slits in it. You set up the barrier so that the only way that any light can reach the photographic plate if it goes through the slits.
You then shine the light towards the barrier, and thus 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 is consistent with the claim that particles can exist in multiple 'life courses' until they are observed, but 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 its own other possible existences. Yes, it's weird. But it appears to be true, or as close to true as we can currently conceive, and it seems to be representative of how all small particles work. For a more technical description, see Young's Slits
This experiment has been redone many many times, including with a number of interesting variations. For example, it has been repeated using carbon buckyball molecules, and they were found to work the same way as photons; they travel as waves, show interference patterns, and arrive as particles (but in the locations that would be expected from an interference pattern). But because buckyballs are much larger than photons, they are also much easier to measure. In one experiment, detectors using photons to detect the path of the buckyballs were placed at each slit; although nothing else was changed, the buckyballs did not travel through both slits (as they do when not measured); instead each buckyball traveled through only one slit, and no longer arrived at the screen in an interference pattern, instead arriving in two patches behind the slits (just as traditional Newtonian physics would have expected). Observing them at the slit 'made them choose' which slit they went through.
This experiment was carried further: this time, the location of the buckyballs were not measured at the slits, but just before hitting the plate. This experiment also found that observing the position of the particles resulted in them hitting the plate without an interference pattern, as if they had traveled as particles along their entire path. This suggests that because they would be observed later, the buckyballs 'chose' to not travel as waves. This highly counter-intuitive experiment is known as Wheeler's delayed choice experiment.
You can find even more weirdness using quantum erasers and delayed choice quantum erasers.