Hundreds.

If you're talking about people, I'd like to say one: the human race.

If you are referring to scientific classification, last I heard it was standing at only four. http://www.racialcompact.com/racesofhumanity.html However, I do not agree with this. First, the term "race" is incorrect. There are no terms that, by definition, do the job but the closest, to me, is "ethnicity" so I use that. Race, or ethnicity, is defined by genetics. I say that any group that is distinct enough to have genetic markers of its own, defining the individual as such and so race/ethnicity, should meet the scientific classification of "race." That list would be very difficult to put together and is much, much more than four. My people are listed, I think, as Caucasoid because Punjab Indians are listed as Caucasoids, some debate as to if we are Mongoloid ("Asian") because there is a strong debate for us not being Punjabi, rather coming THROUGH India from Asia (I believe this theory, myself). However, we have a very distinct genetic marker on our Y chromosome that is ONLY found in my people. Last I heard, genetic studies have progressed to possibly having isolated our mitochondrial markers as well. Genetic studies on us started relatively recently. I am certain that a great number of peoples listed under the big 4 are the same, with the same degree of genetic identification.

Whichever number of races you like partitioning the human race into. Including ONE. The number of eye colors, nose shapes, ear shapes, blood types, hair colors, eyebrow thicknesses, etc, etc, etc that you can use as partitioning lines are near infinite.