What did the Romans call black people (sub-Saharan)? In classic Latin as well as vulgar latin and slang.
Ancient Romans used the word Aethiops/Aethiopem which was derived from Greek Αἰθίοψ (Aithíops).
But the other answer isn't right when it says that the word Aithiops had no meaning except for the land of "Ethiopia". Instead, the word is a combination created from αἴθω (aíthō, “burn”) + ὤψ (ṓps, “face”).
So the Greco-Roman word for the blacks was basically a "burned face". The name of "Ethiopia" was derived from "burned face". In the Greco-Roman world, "Ethiopia" was believed to include the whole non-Roman Africa. It was the universal word for that vaguely known part of the world.
The idea that old nations didn't recognize races is absolutely preposterous from a historical viewpoint. On the contrary, it's just the last 30 years when it became fashionable for some people to deny races and their importance. This denial has absolutely no precedent in any historical epoch.
The original meaning of Aethiops was purely racial. This is demonstrated by dozens of proverbs and important texts. For example, the Bible's Jeremiah 13:23
si mutare potest Aethiops pellen suam aut pardus...
which is currently translated as
Can the Ethiopian change his skin, or leopard its spots?
That's the modern translation despite the fact that the context makes it very clear that by definition, "Aethiops" was someone defined by having a dark skin. For thousands of years, "whitening the Aethiops" was considered the #1 idiom for "doing an impossible thing". One can't whiten a black because blacks are defined by their not being white.
In my earlier answers on similar questions,
I expressed the opinion that neither Romans nor Greeks nor Jews (of the time when the Bible was written) had words to designate races. Race is a modern invention. I challenged anyone to find a counterexample. No counterexample was ever shown to me. People were described by their country of origin, religion, ethnicity, but not by race. From the ancient literature we cannot even make a conclusion of the skin color of various personages. Was the Queen of Sheba who visited Solomon black? How about personages of Heliodorus's Aethyopica? Who of them was black?
Besides Scipio Africanus, there was an emperor Gaius Pescennius Niger Iustus, a 100% Italian, of a noble Roman family. Of course in Latin "niger" means "black". But this had nothing to do with race.
EDIT. In his answer, Lubos Motl claims that the word "Ethiopians" in the ancient literature describes a race. This is not so. Nubians are of the same race, they are frequently mentioned in the ancient literature, and there is no word which would mean "Nubians and Ethiopians and other similarly looking people".
Or take Numidians. Can someone tell me of what race and what color were Numidians, so frequently mentioned in the Roman history? In any case, I am quite certain they were never called "Ethiopians":-)
The cites from the Bible given in his answer do not have racial character: they only mean that Ethiopians have dark skin, and this cannot be changed. Which is simply a statement of fact. The Bible has many statements discriminating various peoples, but ALWAYS by language or religion. And never by race, as far as I now.
In the comments, people mention Helens/Romans vs Barbarians. But what does this have to do with race/skin color?? For the Greeks, Barbarians were Persians/Medes, first of all, people of "Arian race". For the Romans, Brits, Germans and Gauls (before Britain and Gaul were conquered) were Barbarians. Is not it clear that "Barbarians" has nothing to do with race?