The "color" of a diamond refers to its degree of "yellowness."  The ideal diamond is completely colorless, and therefore it will be the most expensive.  The Gemological Institute of America (G.I.A.) grades color alphabetically from D (totally colorless) to Z (yellow):

For a diamond to be considered "colorless," the G.I.A. requires that it be a D, E, or F.  However, the D-Z scale is continuous, so the difference between an F and G is very small.  The average color for engagement diamonds in the United States is G to H. 

Jewelers have two tools at their disposal to judge the color of a given diamond.  The first is what's known as a "reference set" of stones.   A jeweler will compare the stone in question with a set of stones of known color (the set is typically made of cubic zirconium!), and make a qualitative determination as to the color grading of the stone in question.  The second, more precise method, is to use a colorimeter, which is nothing more than an electrical device that will measure the optical characteristics of the stone and report the color to within 1/3 of a grade.   Be aware that most jewelers routinely "round up" the results of a color test.  Therefore, a stone that is only slightly better than an F grade automatically becomes an E (and hence becomes more expensive for the consumer to purchase).

When judging the color of a diamond, it is crucial to see the diamond unmounted.  Ask to see the printout from the colorimeter, or ask the jeweler for a reference set of stones to make the comparisons yourself.  To do this, place the diamond in question next to the reference stones face down on a white piece of paper, and compare the color of the stones until you get the best match.

Perhaps the most important factor to consider when selecting color is the type of setting you plan on using.  If you plan on mounting the stone on a platinum or white gold setting, consider a diamond in the D-G range.  Yellow gold will be much more forgiving to a less than colorless stone, but regardless of the setting, the diamond will start to appear yellow if the color grade is lower than about J.