Whilst it's actually a myth that most people in the Middle Ages thought that the world was flat, and that Columbus thought he was going to sail off the edge (why would anybody set off in the first place if they thought that?), the modern combination of the internet, too much spare time and a failing education system has led to a resurgence in the ideas amongst the foil hatted folks of online forums.
But, it's not fair to attack the theory through ad hominem attacks on its supporters, no matter how tin foiled their hats are, it's the ideas you have to tackle if you want to get to the bottom of the issue. Luckily enough, the major arguments behind the theory are so poorly thought through, easily falsifiable and often just plain wrong, that refuting the ideas is just as much fun as impugning the supporters.
Flat Earthers, as you may well have gathered, believe that the Earth is flat, and they've apparently got the science to prove it. To the uninitiated, a lot of this proof does seem to consist of, "Well, it looks flat...", but given the fact that even celebrities are now getting on board with the idea there have got to be some better arguments, right? Right?