According to this article from the BBC, the flat-earth community is alive and well these days in the 21st century. It's the 50th anniversary of the founding of NASA, an event that has sparked renewed interest in conspiracy theories about faked moon landings and fake, spherical pictures of the earth taken from space. One notable organization is the Flat Earth Society, whose motto "Deprogramming the masses since 1547" is almost too funny for words. (Check out their site for a good laugh/cry).
The flat earthers, as a group, deny that our planet is a sphere - that's clear and simple. The specific shape of the earth, however, is still a topic of debate. Some think the earth is a disc with infinite horizontal distance. Others think it a disc with a fixed diameter, like a giant coin.
Most interestingly, the article points out that flat-earth conspiracies were not pervasive during the Dark Ages, as is the common understanding, and only received widespread attention in the 19th century, as Christian authorities sought to counter the progression of scientific rationalism. Faced with the popularizing of naturalistic accounts of earth history, Christian "thinkers" toured the country spouting alternative accounts of the organization our planet and solar system.
This is a perfect example of misguided skepticism. Instead of questioning complex, fantastic explanations for phenomena, replacing them with parsimonious, evidence-based answers, the flat earthers simply deny the evidence - claiming it's widespread propaganda from the Authority - and insert an alternative explanation.
Skepticism is vital for scientific progress and critical, rational thinking in everyday life. Question everything, sure, but allow yourself to adjust that skepticism according to the evidence presented to you.