The exact number is infinite, but a finite number exists in terms of colour grades.
For 'white' or 'colourless' diamonds, the grades begin with the letter D -- colourless -- and trail off to the letter Z -- light colour, with five broad categories of grades.
For coloured diamonds, please review the Color Grading system, also linked below, to identify the hue, tone and saturation variables available for grading coloured diamonds.
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There's an infinite amount of colors.
Innumerable, really. Eighty percent of all diamonds mined are industrial diamonds, with a separate classification for valuation, including colour. Only 20% of all diamonds a gemstones, with a named colour. That said, because diamonds are natural, there are as many colours as Mother Nature provides. You can review the chart for 'white' diamond colour, below. As well, you can read the names of 27 types of coloured diamonds, below.
There are virtually an infinite amount of different skin colours that exist since the possibilities of shades are infinite.
Many: yellow, orange, blue, pink, white, black, red, green and all tints. The fancy colors are most expensive among naturally-mined diamonds because they are extremely rare. Very rare fancy colors are now can be grown in a lab and their price is several times less than natural counterparts.
There are numerous 'kinds' of diamonds -- too many to list. The number of diamonds cannot be determined, since not all diamonds have been mined to date.