Given the fact that all modern digital cameras have an LCD screen screen on the back that will actually display the image, what is the point in the histogram?
What is the histogram showing us?
In short, it is a graph showing the number of pixels with a particular brightness. Brightness is divided into 256 increments from values 0 (left; full black) to 255 (right; full white), with everything in-between being, well 'greys'!
How is it useful?
the screen on the back of your camera sucks and is not a reliable method to determine if you have the right exposure. I have, many times, taken pictures looked at the LCD screen, decided it was too dark, so increased the exposure and taken it again, only to discover it was dark because i had my sunglasses on. The histogram will show you the exposure of your image, so you can make a better judgment as to if it is correct.
So a perfect exposure has a perfect histogram?
No. There is no such thing as a perfect histogram in general, as different scenes have different requirements for exposure and will result in different histograms. A particular histogram will be perfect for one scene but NOT another. For example, a portrait with a white background will have a histogram skewed to the right, where as a portrait with a dark background will have a histogram skewed to the left. This is irrespective of the subject being perfectly exposed. So for each situation the histogram is perfect, but this perfect histogram is not universal! The three images on the left all have the exposure and look I wanted, but the histograms are all very different! :)
How to use histograms to their best.
Side note: The range of the histogram is dependent on the quality of your camera sensor and is called the dynamic range. The larger the dynamic range of your sensor, the more detail it can detect in the highlights and shadow areas.
Clipping, and how to avoid it. Clipping is basically missing information. Pixels with values 0 and 255 are 'unrecoverable' pixels, no matter what you do to your image, these pixels will be pure black (0) or pure white (255). So if you have an image with a lot of pixels at either of these extremes of the histogram, you are missing information in the corresponding areas of the image. In some situation this is unavoidable or even desired, but in others you can 'shift' the histogram more towards the midtowns in order to capture as much information as possible in the scene by using exposure compensation or manually changing your camera settings.
The bottom line
Histograms tell you how much information you are NOT capturing and will allow you to decide if that is ok or not! :)
The only rule I tend to use is to avoid clipping, but then again, sometimes I even ignore this rule (the black and white picture for example)! :)