A difficulty in answering this question is that the French perspective on race is quite different from the North American perspective; in particular, referring to someone by their skin color (as opposed to their nationality, or their national origin) is generally more offensive, or at least less politically acceptable, in France than in North America — even when the reference is purely descriptive. The more politically correct terms tend to veer towards national origin or ethnicity rather than skin color.
For this reason, terms that might be considered simply descriptive of skin color in North America may be considered more offensive in a similar context in France.
This is a particularly acute problem to describe "whites" in France — to my knowledge, there is no widely accepted politically correct term for it.
With that said —
Descriptive: noir, à la peau noire, de teint noir, mélanoderme (technical term to mean having black skin, not used very much)
for Africans: d'origine africaine, d'origine sub-saharienne
for African-Americans: noir américain
for Afro-Caribbeans: countries are used, e.g. haïtien, dominicain, jamaïcain, cubain, etc.; or (for DOM), guadeloupéen, martiniquais.
Slang: renoi, black, kebla
Offensive: nègre, blackos
This group is more present in France than in the US and has therefore more terms to describe it.
Descriptive: de teint mat, de peau mate, de teint olive (this just describes skin color, and may cover other ethnicities as well).
Politically correct: d'origine maghrébine, d'origine nord-africaine
Common: arabe (even if many are from the non-Arabic berber people), oriental (even if most people from North African origin in France come from places that aren't particularly much to the East of France).
Slang: rebeu, blédard
Descriptive: asiatique, xanthoderme (technical, not used much)
Politically correct: d'origine asiatique
Offensive: jaune, niakoué, chintok
Descriptive: blanc, de teint blanc, de peau blanche, leucoderme (technical, not used much)
Politically correct: as mentioned above, there isn't really a good word here. You kind of have to choose who to offend. For a while, de souche européenne was used for this, and shortened to français de souche, but has become associated with far-right nationalism. In the other direction, the derivative souchien was created, and is understood as pejorative for whites when read as sous-chien.
It may be easier to describe non-whites. A euphemism for this is issu de l'immigration, even if many immigrants to France are whites and many non-whites have been French for many generations (e.g. martiniquais, guadeloupéens). Another euphemism for non-whites is issu d'une minorité visible.
Slang: babtou, toubab, gaulois
Offensive: fromage, jambon, sous-chien