The Flat Earth theory is the mistaken idea that medieval Europeans believed that the Earth was flat based on a literal reading of the Bible. They tried to convince Christopher Columbus that he would fall off the edge of the Earth, the story goes, but Columbus proved that the Earth was round by discovering America. 
In fact, the ancient Greeks knew that the Earth was round, and had correctly estimated its diameter. Medieval Europeans knew this, and had correctly warned Columbus that he would never get to India with his limited supplies. Columbus relied on a fallacious argument that the Earth was much smaller than the Greeks estimated.