I find that I can read a printed explanation and not be able to get a strong understanding of the concept. I need to see a graphical representation of the concept to really get a feel for it. Depth of field is much easier for me to understand with a nice diagram.
The diagram above shows the basics of depth of field. An object at point y is at the focus plane and will be in sharp focus at the detector. An object at point x is behind the focus plane and will focus in front of the detector. The light rays from x won't just stop at their focus point, they will continue on to the detector where they will form a blurred circle the diameter of c. The opposite will happen to an object at point z. Since it will come into focus behind the detector it will also form a blurred circle the diameter of c.
If we define c as the amount of blur your eye tolerates in an image, c is the Circle of Confusion. The distance between object x and object z is now the Depth of Field (DOF). An important point to remember is that not everything within the DOF is equally sharp. The object at the focus plane is sharpest and objects will get progressively fuzzier as you move either forward or backward. The fuzziness increases until it becomes visible at the limits of the DOF. Another point to remember is the DOF is only valid under a certain image viewing size and distance. If I were to get up close and personal with the image (zoomed up to full size on a computer screen), objects the were sharp previously may now become fuzzy as I have now changed the image magnification and thus changed the CoC.
Next: How the aperture makes the DOF larger or smaller.