I would like to know how to address the difference between them and how relates to each other.
The focal plane is the distance from the camera at which the sharpest focus is attained. Depth of field is the range of distances either side at which the focus is "acceptably sharp".
The "acceptably sharp" part is what causes confusion as you need to make assumptions about viewing distance and visual acuity. For this reason a standard figure for the amount of blur that is acceptable tends to be used. It's thus important to realise that any depth of field numbers are approximate, suitable for comparison (i.e. this setup gives DOF of X meters, this setup gives DOF of Y meters) but the actual range that appears sharp will likely differ.
For lenses which exhibit a defect known as field curvature, the focal plane wont be flat but will curve outward, meaning when photographing something dead flat you could end up with the centre sharp but the corners out of focus.
Tilt-shift lenses allow you to tilt the focal plane relative the camera to better suit the subject (for example tilting the focal plane down would help you get more of the ground in focus).
There are two focal planes, front and rear. To quote Wikipedia: The front and rear (or back) focal planes are defined as the planes, perpendicular to the optic axis, which pass through the front and rear focal points.
So, what that means is that everything at the focal plane will be perfectly in focus. That's a simplification, lens optics are not so perfect that the entire field of view across that plane will be perfectly in focus, but it's close enough for discussion.
Depth of field is the range of distance from a point behind the focal plane to a point in front of the focal plane for which the image is sufficiently in focus so as to be seen as sharp to our eyes. This entirely varies based on a number of factors:
Aperture of the lens. The smaller the aperture, the greater the depth of field. Wider aperture, means more shallow.
Focal length. Shorter focal lengths tend to have greater depth of field. Telephoto lenses compress and isolate the subject with shallower depth of field. Dedicated macro lenses (which are often short telephoto anyways) further effect this by allowing extreme close focussing creating razor thin depth of field.
Sensor size. Larger sensors tend to have shallower depth of field for the same aperture and focal length.
For the most part, in Photography, you're really more interested in the depth of field.