What was Young's Double Slit Experiment, and why is it important?

Thomas Young's Double Slit Experiment could well be the most important experiment in the history of science. At the time its significance was not really appreciated, but about a century later, adaptations of the Double Slit Experiment laid the foundation for quantum mechanics, which now underlies basically all of physics as we know it (except astrophysics, because we've not yet been able to make sense of gravity in terms of quantum mechanics).

The experiment is actually quite simple: You have a light source in a dark room, and you put an opaque screen in front of it, and then some film or another light-recording apparatus behind that. You cut two very thin slits in the opaque screen that are very close together, and then you observe what sort of pattern emerges on the film. Then you try covering one of the slits and compare the new pattern.

The question Young sought to answer was whether light is a wave or a particle (the correct answer is both... sort of. Or maybe neither?).

If light is a particle, then the light coming out of the two slits should be additive; the pattern with two slits should be basically the same as the pattern with one slit, only twice as bright (and partly shifted, based on the distance between the two slits).

But if light is a wave, then the light coming out of the two slits should exhibit interference; the waves should overlap in such a way that they add in some places and subtract in others. The resulting pattern should have dark parts and light parts, dark where the waves canceled out and bright where they added together; these should alternate in proportion to the wavelength.

What Young observed was the latter: There were dark parts and light parts, just as you'd expect if light is a wave. Young and his contemporaries left it there: Light is a wave!

But later expansions of his experiment showed that this wasn't quite right. Once we made more precise equipment so that you could measure whether a particular photon went through one slit or the other, we found that the pattern looked like particles! Trying to understand why we sometimes got a wave-like pattern and other times we got a particle-like pattern is what led to quantum mechanics.