The Americas are named for one Amerigo Vespucci, an Italian cartographer who was one of the first to put them on European maps.

Asia is named for the Roman province of Asia, which was roughly the area of modern Turkey. EDIT: it also appears to have been used by Herodotus well before this.

Africa shares a similar origin, with the Roman province of Africa, based around what's now Tunisia and Libya.

Europe appears to be named for the classical mythological figure Europa, but I can't speak for why that is, unfortunately. If someone else does, please chime in.

Australia is a shortened form of the Latin Terra Incognita Australis, meaning the unknown southern land, if my Latin holds up. Old maps frequently have much of the Southern Hemisphere labeled as such, as Europeans hadn't explored so far south. The name stuck with the modern land around the time the British began colonizing it.

Antarctica is just Latin for something along the lines of "the opposite of the Arctic". Think of it as Anti-Arctic, and it becomes a little clearer.