Much fuss is made about quantum things being both waves and particles. This can be illuminated by the famous double-slit experiment. The nature of the experiment can be illustrated by using some everyday waves and particles before we delve into any quantum jiggery-pokery.
The setup of the experiment is like this: from one end of the room you're going to fire your waves or particles. These are going to hit a barrier with two vertical slits in it and then you're going to look at the results on the wall behind that barrier.
First we're going to make some particles and we'll use bullets for that. If we fire bullets at the two slits, you'll see a pattern on the back wall that matches the slits; a vertical scatter pattern in two areas that correspond to the slits in the barrier (as in the left picture, below).
Next we're going to make some waves. I'm told Mexicans are good for that but we'll see things a bit better if we use some water instead. As our water wave hits the two slits it'll spread in concentric waves radiating out from each slit. The two subsequent waves will then interfere with one another. Where a peak and a trough meet you'll get flat water, where two troughs meet you'll get a negative (lower than flat water) wave and where two peaks meet you get a positive (higher than flat water) wave. You'll see this interference pattern on the back wall (as in the right picture, below).
This is what we'd expect to see from real-world wave and particle systems. Now let's try the same experiment with something quantum, like light. So let's fire a source of light at the slits and see what we get. It looks like light is behaving like a wave and we get the interference pattern like the picture on the left, below.
But light is also supposed to be a particle (called a photon), so what happens if we slow the light source down so it's firing a single photon at a time. We should expect to see a particle-like pattern on the back wall like we did with the bullets, but we don't. We see a wave-like pattern even firing a single photon at a time. The single photon must be going through both slits at once or we wouldn't see the interference pattern. Each photon is interfering with itself (below, right).
That's all a bit mind-bending so let's try and catch the light out by putting a detector next to the slits to really see which slit the individual photons are going through. The light won't be caught out, though. If we start watching what it's doing, it'll start to behave like a particle and we'll get the bullet-like pattern on the back wall.
There's another peculiarity here too. What if we release the individual photons and then, only after they've been released, turn the detector on? Surely at the point of release the photon has already made a decision to behave as either a wave or a particle. Alas not, we can't catch light out like that either. If the detector's on, no matter when we switch it on, it behaves like a particle; if the detector is off, it behaves like a wave.
Make of that what you will but it perhaps illustrates how the observer has an intrinsic role to play in quantum systems.
So what is light — a wave or a particle? It's probably best to see as something that can exhibit either wave-like or particle-like behaviour depending on how you observe it. It might be that, deep down, it's neither a wave nor a particle, or maybe it's both. In my mind's eye, I see it as a field with fluctuating values at each point in space but I've no idea if that's what it actually is. Quantum mechanics tells us much about what we can observe — what we can know about the universe — and that makes it a theory of information in essence. What is actually happening at the fundamental level is less clear.
NB. Apologies for my crappy artwork — I'm terrible at drawing.