As of the latest Census, there were 19 categories, and people could choose more than one -- something that wasn't an option before the 2000 census, but has become increasingly common. Looking back at how our definitions of race have changed, it's not a stretch to imagine people in 2110 looking back at our categories and finding them odd (or even offensive).
If you're interested in digging into that history, the Pew Research Center created a cool infographic showing race and ethnicity categories over the years. It's an interesting look at how Americans and our government perceived race through our history. On the left is past years, which you can change, and on the right are the 19 categories in the 2010 census.