So most times, when something is discovered, the person who did the discovering has the privilege of naming it. And often times, they choose to name it after themselves.

AMERICA: These continents were named after an Italian explorer named Amerigo(Americus) Vespucci. He proposed that the newly discovered lands were a new continent entirely and not part of India. Then in 1507, a cartographer named Martin Waldseemüller drew a map and labelled the newly discover lands as America meaning land of Americus. Everyone loved it so it stuck.
EUROPE: This continent was originally called Europa and before that Europs by the Greeks. The Greeks believed that Zeus impregnated a Phoenician princess and she was taken to Crete. There she bore three sons. Either way, its origins are hazy at best. Europe means wide face.
AFRICA: This comes from the Romans marking out a place called Carthage in what is now Tunisia. It means land of the Afers. It only referred to northern Africa originally, seeing as all they saw was desert and no one wanted to venture through it to find out if something truly laid farther down. Eventually, as time passed, it began to be used to refer to the entire continent because explorers found that there was more farther south than they originally thought.
ANTARCTICA: This was named by the Greeks. It was originally called anti artikos meaning opposite of the arctic. But the arctic was named after the constellation Ursa Major which was a bear. arktikos meaning bear. So anti artikos actually means opposite of the bear. The word Antarctica gained prominence when John George Bartholomew, a Scottish carthographer started using it in 1890.
ASIA: This has Greek origins. It was used to describe the lands which were then the Persian empire and Anatolia(Turkey). As time passed and the enormity of Asia was discovered, Anatolia was called Asia minor, or smaller Asia
AUSTRALIA: This continent was named by English explorer Matthew Flinders.. The Dutch had sent navigators who had already explored and charted the western, southern and northern coasts of the continent in the 1700's. They called it New Holland but Flinders name replaced it shortly after.