Africa comes from the Roman designation of the area around Carthage in what’s modern day Tunisia, Africa Terra, meaning “land of the Afer” (pl. Afri). It only began to represent Northern Africa (excluding Egypt) after the administrative restructuring of Diocletian in 293 and later the whole continent as knowledge of it grew in the middle ages. The etymology of the word is unclear and but could be Phoenician in origin.
America comes from the name of the Italian explorer Amerigo (Americus) Vespucci, the first to propose the newly discovered lands were a new continent and not India. Cartographer Martin Waldseemüller named the continent America (land of Americus) in his 1507 map and the name stuck.
Antarctica comes from the Greek anti arktikos, meaning “opposite of the Artic”, which in turn is named for the northern constellation Ursa Major, the great bear. Arktos is bear in Greek. It’s use as the name of tyhe continent is attributed to Scottish cartographer John George Bartholomew in the 1890s.
Asia is Greek in origin, its first recorded use being by Herodotus around 440 BCE to describe the lands of Anatolia (modern day Turkey) and the Persian Empire. The name probably comes from the Hititte name Assuwa, which was used for the Aegean coast of Anatolia in 1400 BCE. As the Asia stretched to cover the lands further east, Anatolia started to be called Asia Minor (the lesser or small Asia).
Europe comes from the Latin Europa and it from the Greek Europs, meaning “wide face” although the ultimate origin is uncertain. In Greek mythology it was the daughter of a Phoenician king, seduced by the god Zeus and taken to Crete, where she bore three sons of him. Europe as a geographical term was first recorded as used by Greek geographers Anaximander and Hecateus in the 6th century BCE, although it was used before by the Greeks to denote the lands inhabited by them.
Oceania derives from ocean, it in turn from the Greek Okeanos, the great sea surrounding the world. It was first used by Danish geographer Conrad Malte-Braun around 1812 to denote the small Pacific islands, with Australia and New Zealand added later.