An understanding of color and the way it behaves (or sometimes misbehaves) is crucial to successful painting. Here, I'll explain warm and cool colors and, more importantly, color influence, or bias. Knowledge of the bias specific colors possess will mean less frustration in mixing the desired color, either vibrant or muted.

This discussion assumes that you are familiar with basic color terminology: Primary, secondary, tertiary (sometimes called intermediate colors), complementary, or opposite, colors, etc.

Let's get started

Simply speaking, warm colors are those that fall within the yellow/orange/red portion of the color wheel and cool colors are those that fall within the violet/blue/green portion. A color's relative warmth or coolness is known as temperature. Temperature, in this context refers to the perception of a color ~ when you see blue, you typically think "cold", when you see red, you typically think "hot". It also refers to how certain colors appear when placed in the vicinity of another. For example, place violet next to red, and the violet appears cool. Place violet next to blue, and the violet appears warm.
To carry this concept further, within the individual warm/cool color ranges are additional warm and cool colors. By this I mean that although a particular color falls in the warm or cool range, that color family itself will contain warm & cool versions of that color family.
A good example of this is found in the "warm" red family. Cadmium Red Light (PR 108), which leans towards orange (orange bias), is a warm red. Quinacridone Crimson (PR 122), which contains some violet (violet bias), is a cool red.
If you paint these two reds side by side the temperature of each becomes apparent. When you isolate a color family, red in this instance, the different reds in that family will appear warm or cool relative to each other.

It is at this point where I cease referring to a color by its temperature, but rather, by its bias, or influence. There can be some confusion when it comes to describing color by temperature alone, particularly violets and greens. It is much easier to think of temperature in the general sense of color families, and apply bias to specific colors within those families. In other words, "warm" red family: Cadmium Red Light = Orange-red (red with an orange bias), and Quinacridone Red = Violet-red (red with a violet bias).
From here on out, I will be referring to a color by its bias, rather than by temperature in the examples that follow. "Warm" and "cool" will be used in the broader sense of color families, not specific colors.

Bias,or influence, refers to specific properties of a hue according to its location on the color wheel. Except in light waves, there are no perfect, "pure" primary colors (red, yellow, blue). In the case of physical color, each "primary" pigment will carry small amounts of other colors that influence the overall appearance.
Red pigments carry varying amounts of orange and violet with trace amounts of green (usually not enough for concern)
Blue pigments carry varying amounts of violet and green with trace amounts of orange (usually not enough for concern)
Yellow pigments carry varying amounts of green and orange with trace amounts of violet (usually not enough for concern).
Each "primary" will have a different bias, depending on the amount of contaminating colors present. So, instead of three primary colors, your palette should contain six "primary" colors in order to be able to mix vibrant or muted secondary colors as demonstrated later.
So, you will need:

• Two reds: Orange-red (a red with a slight amount of orange present) and Violet-red (a red with a slight amount of violet present)
• Two blues: Green-blue (a blue with a slight amount of green present) and Violet-blue (a blue with a slight amount of violet present)
• Two yellows: Green-yellow (a yellow with a slight amount of green present) and Orange-yellow (a yellow with a slight amount of orange present).

There are several pigments to choose from that fall within each of these ranges. The color wheel below shows some examples of six "primaries" painted side by side to illustrate this. I've also listed some other pigments that fall within the ranges. These will yeild slightly different results when used in mixes due to stronger or weaker biases. Once you understand the concepts that will be demonstrated, you will see that, with minor adjustments, you'll be able to choose any color within a range based on its bias and achieve the mix you need.