Oz, it goes like this:
On the backside of the screen of your monitor is a tight maze of small lightsensitive dots. These come in groups of three, a Red, a Green and a Blue one, and are called phosphor triplets. Give them electricity and they light up.
Now electricity can have an infinite number between zero and maximum, so your monitor can display a nearly infinite number of colours.
But your comp must tell your monitor where it wants which intensity of R,G andB to make one colour dot. This asks a lot of calculating power.
The easiest is simply on and off. But that's not much, is it?
So little by little they came to 256 values, starting with 0 (off) and 255 (max) and 254 in between. This number is not chosen at random. Because a comp only can count with zero and one (two values), you get 256 when you give 8 bits to each colour.
from 00000000 to 11111111 going by
00000001,00000011,10010011 etc etc. Try it out is you want you get 256 possibilities.
Or just take my word for it.
Practically speaking this means that and Red, and Green and Blue each have 256 possible different settings of intensity, so added together this gives you the 16 etc million colours.
But nowadays processors and video cards are so powerful that they can double this rate and give 16 bits per "channel". This will be a visible improvement as there will be an awful lot more hues that can be displayed, and, for example, gradients won't show banding anymore.
But you need a powerful comp like all new machines are, with a 32MB video card.
To get an impression why Stu and Mark (and me) are so disillusioned about this, set your monitor to display only 256 colours. You'll see the difference. And 8 to 16 is yet again a big improvement.