The double-slit experiment is a cornerstone example in quantum mechanics that neatly ties together concepts from superposition, interference, uncertainty, measurement and quantization.
When light passes through a narrow object, like a thin slit, it spreads out. When we place two thin slits side-by-side, the light passing through one spreads into the light from the other. By looking at the light far from the slits, we can see regions of constructive interference where light is more intense, and regions of destructive interference where there is no light at all. We can understand this as the light waves that spread from the left slit interfering with the light waves spreading from the right slit.

But from quantization, we know that light is emitted in discrete chunks called photons. What happens if we send a single photon into the double-slit experiment? If we measure which slit the photon goes through by placing a photon detector right after each slit, we see that it only goes through one slit or the other. But if we measure far away from the slits, we see an interference pattern that builds up slowly, one photon at a time:

Video Source: P. Kolenderski et al. (IQC/UWaterloo), Scientific Reports 2014; 4:4685. (Jennewein lab)
These are the two golden rules of quantum mechanics in action. The photon went through a superposition of both slits, and the wavefunction from the left slit interfered with the wavefunction from the right slit. Unlike the classical double-slit experiment, it is not the wave energies that interfere, but rather the wavefunction probabilities.
However, if we measure which slit the photon goes through, we see it in one slit or the other, never in both at the same time. Measuring which slit the photon goes through disturbs the superposition and destroys the interference. Indeed, precisely knowing which slit the photon went through and which interference pattern we see at the same time is impossible, an example of the uncertainty principle in action!