Nano Nook for another example image.
So what is quantum tunneling? Well, think back to the potential that goes into Schrodinger's equation. Speciffically, I want you to consider a potential that looks like this.
You can visualize this like two hills that are sitting next to each other. Image you're sitting in the valley on the left, but you haven't eaten all day an you're exhausted. All your friends are waiting in the valley to the right, but you can't get there! You don't have enough energy! Well, this is the classical physics way of looking at things - no matter how hard you try, you don't have enough energy to make it up and over the hill. Well, quantum mechanics says, you don't have to! Since your wave function extends in space, a small part of it overlaps with the other valley! What does this mean? There's a finite chance that you could be in the hill to the right! This is what quantum tunneling says, and this effect has been measured and is currently used a device called the scanning tunneling microscope (as well as MANY MANY other technologies.) Here's a gif I stole from wikipedia that shows this happening... sadly I didn't have time to make a simulation of my own.
Here you see a quantum particle penetrating a wall.
So how does the scanning tunneling microscope work? It's actually reasonably complicated, but the basic principle is as follows. A tungsten tip is held above the surface that you wish to image. There is nothing but vacuum between the tip and the surface, so normally, there's nothing to allows the electrons to flow to the surface. Fortunately for us, there's a finite probability that the electrons will "tunnel" to the surface, and this effect allows us to image the surface to extreme detail. Pretty cool, huh?