Today it is well known that the Earth is a sphere, or very close to one (its equator bulges out a bit because of the Earth's rotation). When Christopher Columbus proposed to reach India by sailing west from Spain, he too knew that the Earth was round. India was the source of precious spices and other rare goods, but reaching it by sailing east was difficult, because Africa blocked the way. On a round globe, however, it should also be possible to reach India by sailing west, and this Columbus proposed to do (he wasn't the first one to suggest this---see below).
Sometimes the claim is made that those who opposed Columbus thought the Earth was flat, but that wasn't the case at all. Even in ancient times sailors knew that the Earth was round and scientists not only suspected it was a sphere, but even estimated its size.
If you stand on the seashore and watch a ship sailing away, it will gradually disappear from view. But the reason cannot be the distance: if a hill or tower are nearby, and you climb to the top after the ship has completely disappeared, it becomes visible again. Furthermore, if on the shore you watch carefully the way the ship disappears from view, you will notice that the hull vanishes first, while the masts and sails (or the bridge and smokestack) disappear last. It is as if the ship was dropping behind a hill, which in a way is exactly the case, the “hill" being the curve of the Earth's surface.
To find out how the distance to the horizon is calculated, please see the chapter “Distance to the Horizon".