Depth of Field--sounds very exotic. Well, kind of exotic. Maybe.
It's simply this: the depth of the area of the shot that's in focus, given as a distance like meters or feet.
Depth of Field (or DOF from now on to spare my poor keyboard) is largely determined by the aperture (or f-number) of the camera, but the focal length (zoom, or mm of the lens) and distance to the subject also play a role.
It's possible to get an equally narrow DOF with a 50mm lens at f/1.8 as it is with a 300mm lens at f/5.6.
The near plane is the imaginary line (er, plane) where things start being in focus for a given DOF. The far plane is on the far side of the subject, and everything past it will be out of focus again. Everything between the planes is in focus.
The hyperfocal distance is determined by your focal length and your aperture. If the subject is past the hyperfocal distance, everything to the horizon will be in focus. This is how those fixed "focus to infinity" cameras work. Those often won't focus nearer than two meters (6 feet) or so... that's where the near plane is, while the far plane is at infinity!
Click and drag the lens dials on following toy for a bit to see how it works. Everything between the lines around the subject (which represent the near and far planes) will be in focus in the final shot, while everything else will be out of focus to varying degrees. Finally, notice how the far plane goes out to infinity when the subject moves out past the hyperfocal distance.