The story goes something like this: Back in the Middle Ages, people in Europe were stupid and ignorant, so they believed the world to be flat. Then along came the great Italian visionary Christopher Columbus who realised that the Earth was in fact round. He sought to prove this by finding a westward sea-route to India, but in their ignorance and superstition, none of the monarchs of Europe wanted to fund his venture for fear that he would fall off the edge of the Earth. Then finally Columbus was able to persuade King Ferdinand and Queen Isabella of Spain, and sailed off to prove the naysayers fools.
Almost everything about this story is wrong. First of all, Columbus’s voyage in no way proves the world to be spherical (that’s a better term to use than round, since a flat world might still be a round disk) since Columbus did not in fact succeed in discovering a western route to Asia (in those days the name India was often used generically to refer to Asia). Instead he discovered a continent which was previously unknown to Europeans (modulo a few Norse expeditions), which is perfectly compatible with a flat Earth. The first known circumnavigation of the Earth occurred a few years later from 1519 to 1522 and was led by Ferdinand Magellan and Juan Sebastián Elcano.
So did the Magellan-Elcano expedition prove the Earth to be round? No. That would be about as silly saying that Neil Armstrong proved that the Moon is made or rock rather than cheese. Just as people in the mid 20th century knew that the Moon was a giant rock, so people in the early 16th century knew that the Earth was spherical. The earliest documented claims of a spherical Earth hail from ancient Greece and date back to the 6th century BC. By the 3rd century BC, the roundness of the Earth had not only been definitively established by Hellenistic astronomy, Eratosthenes even performed a remarkably accurate calculation of the Earth’s circumference (250,000 stades, which is between 2 and 20 per cent off from the real figure, depending on what exact definition of stade he used; units of measurement weren’t exactly standardised in those days).
Columbus, however, thought he knew better than Eratosthenes and based his plans on a smaller estimate. Given the true size of the Earth, Columbus could have never made it to Asia. If he hadn’t happened to bump into an unknown continent, he and his men would have run out of supplies and perished at sea long before reaching the shores of Asia. So the monarchs of Europe were perfectly justified in refusing to fund Columbus’s ill-thought-out scheme. Columbus was no great genius; he simply got lucky.