The easiest way to demonstrate the effect of changing depth of field is with some example shots. For some reason I've never been able to explain, every photography textbook I've ever read, and even my university lecturer, used photos of a row of chess pieces to illustrate this, so just to be different I'm going to use my favourite model cars.
This shot was taken using a medium telephoto lens set on a focal length of 43mm, and the front of the Aston Martin DB5 (the silver one in the middle), the point on which the camera is focused, is approximately one metre from the camera. The aperture is set to f4, which is the maximum for this lens, producing the smallest possible depth of field. As you can see only the right front wing of the Aston Martin and the back of the Lotus Esprit (the red one at the front) are in focus. The depth of field for this shot is about 10cm.
The attractive look of the out of focus sections is called bokeh. It's an effect that's particularly desirable for shooting portraits where you want to make your beautiful subject standout from the background. A 'fast' lens with a wide aperture (f4 and lower) is the easiest way of creating this.
By reducing the aperture by two stops to f8, the depth of field is slightly increased, to about 20cm. Now the lettering on the Lotus' tyres is readable, although the classic 1937 Jaguar SS100 (the green one) is still very blurred.
Reducing the aperture to its smallest setting of f22 maximises the depth of field. In this shot everything looks sharp. You can even read the number plate on the Jaguar.