Picture this: you shine a flashlight through a donut in a dark room. On the wall, you'll see a spot of light where the light passes through the donut hole. Add another donut, and you'll see two spots of light. Nothing strange here, right? However...
If you do this same experiment on a smaller scale, you do not get the same results. When the donut holes and the space between them get very small, weird things begin to happen. One hole still makes a round-shaped patch of light, but two holes results in two overlapping sets of concentric circles! This is known as an "interference pattern," and is possible only if light rays traveling multiple paths "bump into" each other along the way.
But what's really strange is that you still get this pattern even if you reduce the amount of light until you're only sending one photon (the smallest amount of light possible) through the donuts. You'll get exactly one photon in exactly one place on the far wall, each time, but if you do that over and over, many many times, you'll see that the individual photons end up grouped into the same exact pattern as when you were using the full flashlight. This means that, somehow, each photon explores every possible path through the donuts, but then seems to coalesce into a single particle just in time to reach the far wall -- we never find a "half-photon."
Some theories say the photon advances through the donuts as a complex mathematical wave pattern, which then "collapses" into a single location (somehow) at the exact moment of observation. (More...)