This is now literary or poetic, so anyone you stumble across using it is probably exposing their erudition. It refers to somebody who is gloomy, morose, irritable or bad-tempered. It came into English in the seventeenth century from Latin atra bilis, black bile. This is the direct Latin equivalent (what linguists call a loan translation) of the Greek word that gave us melancholy, which comes from Greek melas, black, plus khole, bile.
Black bile was one of the four humours of ancient medicine that in their relative proportions determined one’s underlying disposition. The others were blood, phlegm, and yellow bile or choler. These gave rise respectively to the adjectives sanguine (cheerfully optimistic, from the French word sang, blood), phlegmatic (unemotional and stolidly calm), and choleric (bad-tempered or irritable).
However, atrabilious took on some of the idea behind the choleric sort of bile (and one sense of bilious) to mean bad-tempered as well as melancholy.
Incidentally, atrabilious is one of the few words in English that uses this Latin word for black rather than the better-known niger. The most common one is atrocious, which comes from a figurative Latin sense something like “black-hearted”; another is atrium, originally the name of the Roman open hall, so called because the walls became smoke-blackened from the central fire.