Cool/warm generally refers to the position of a color on a color wheel. Half of the color wheel is cool (purple to green) and the other half is warm (green-yellow to red).
By using colors just from either the warm or cool side of the color wheel, you can achieve interesting color harmonies. For example, Claude Monet used a cool analogous color harmony in many of his paintings, using many variations of greens, blues and purples. The result of this is a very calming and peaceful harmony.
The balance between cool and warm in your painting is very important, arguably more so than the actual colors you decide to use. This is because of the contrasting emotions warm and cool colors evoke and the role of warm and cool colors in establishing the overall composition of your painting.
When you are mixing your colors, one question you should always ask yourself is how warm or cool do you want the color to be.
In the video below, you can see this question being asked in a general sense by one of the masters of painting, Richard Schmid:
I would watch that video in full as you get an insight into the mindset of a master painter. It this too warm? Where are my directional lines? Are my focal points interesting? Will this stroke detract from my focal points? These are the questions he is addressing whilst he paints.
Cool/warm can also be used to describe the temperature of a color in relativity to another color. For example, you could have two different blue colors, one french ultramarine blue straight from the tube and another to which you have added some orange. The former will be more warm than the latter. However they are still both on the cool side of the color wheel.