The ancient Greeks first came up with the idea of Antarctica. They knew about the Arctic - named Arktos ("the Bear", from the constellation the great bear) and decided that in order to balance the world, there should be a similar cold Southern landmass that was the same but the opposite ("Ant - Arktos" - opposite The Bear.) They never actually went there, it was just a lucky guess!
In 1773, James Cook crossed the Antarctic circle and circumnavigated Antarctica. Although he did't sight land, deposits of rock seen in icebergs showed that a southern continent exists. He said, "I make bold to declare that the world will derive no benefit from it".
Captain Thaddeus Bellingshausen, a Russian naval officer, was the first to cross the Antarctic circle since Cook. He made the first sighting of the continent, reaching 69° 21'S, 2° 14'W - describing an "icefield covered with small hillocks." on Jan 27, 1820.
For some time, exactly who first saw Antarctica was in dispute. British naval officers William Smith and Edward Bransfield saw Antarctica on Jan 30th of the same year, followed by American sealer Nathaniel Palmer on Nov 16th.
This was the first time a continent had truly been "discovered," meaning that there weren't any native people living there who'd known about it for ages. (info from Coolantartica.com)