Technical Terms:
Hue: The predominant colour
Value: The lightness or darkness of a colour.
Saturation: The degree of colour intensity.
Warm/Cool: Most colours can be associated with a (purely imaginary) temperature. Cool colours tend to blue, warm colours tend to yellow/red.
Mix (Almost) Every Colour with Only Four Pigments!
Red, Blue and Yellow are the primary colours. In theory, you can mix every other colour by mixing the three primaries, adding white to get different tints. Mix red and blue to get violet, blue and yellow to get green etc.
In practice, it is quite difficult to choose the pigments that are capable of giving the widest range of colours. But it isn't a gimmick, in fact I think using just three pigments is an important technique for every artists. Try these colours:
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French Ultramarine
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Cadmium Yellow Pale
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Alizarin Crimson
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Titanium White
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What are the Advantages?
Your colours never get muddy. Using too many pigments in your colour mixes results in muddy, lifeless paint. Most artists agree that you should never use more than three pigments plus white in any given mixture - with this technique it impossible to use more than three.
Mixing Colours is Simpler. Too many other pigments on your palette just confuse the matter. With this technique you just need to ask yourself a few simple questions:
What is the main colour (red, blue, yellow)?
What other colour do I need to add (e.g.. add blue to yellow to get green)?
How much white do I need to add to get the correct value?
OK, how do I mix my colour?
Look at those leaves. They're green. We know that blue and yellow make green, so mix 50:50 french ultra and cadmium yellow. The green we see in the leaves is a little bluer, so add a little more ultramarine. The green we have now is pretty intense ('saturated'), so we add a little white. But somehow that green is still not quite right, its too 'pure' and needs knocking back with the opposite ('complement') of green: Red. So add a little (just a little) alizarin crimson. Success!
Or look at those red bricks. Fine, get a dollop of alizarin crimson. But the bricks are more brown than red, so add a little splodge of yellow. Ooops, we've got orange - that can't be right. Add some white and the mixture looks more promising. Add some blue (the complement of orange) to knock it back a little. That looks better and now we can just adjust the relative amounts of each colour to get the brick colour we need.
What's the Catch?
You can probably only get about 95% of the colours you might conceivably need using these particular pigments. The pigments that would give you 100% probably haven't been invented yet. When I paint with this range of colours I am unable to mix a good substitute for cobalt blue. Neither can I get a very saturated (intense) green.
For general colour mixing hints, see this article.