Indeed. Nothing more than a lucky idiot.
Contrary to popular belief, the idea that the world was round was neither new nor controversial; but the size of the earth was of some debate. In order to get support for his expedition he paid "Experts" to estimate the planet was much smaller than it was - if the Americas weren't there there was no way he would have made it to india as planned - he and the crew would have starved to death long before.
So much Columbus hate here! I'm not going to comment further on that. But I'll comment on this:
The world being round was very much accepted at that time. On of several pieces of evidence is that as a ship goes over the horizon the ships sails disappear last.
And the size of the world was also known, and correctly calculated by the ancient Greeks (Eratosthenes), by measuring the angles of the sun at different locations of the earth (eg Greece and Egypt) at the same time of year. Those Ancient Greeks were really clever.
But Columbus did not repeat that calculation and used the wrong type of "mile" for the measurement leading him to think the Earth was smaller than it is. Kind of like someone thinking a distance is centimeters instead of inches.
Much in these posts is true, except that there's something missing in that of Savlian. Columbus didn't just made a miscalculation in determining the size of the Earth. He stubbornly refused to listen to all the people who told him about Erathostenes' calculations, preferring to believe instead in his own misguided idea that the world must be much smaller than Erathostenes said it was.
The reason that Columbus was one of the first to get to India by crossing the Atlantic (he wasn't the first to try, but, the Norseman aside, he was the first to succeed) was that almost everyone else at the time had the good sense to see that such a journey would be far longer than technology at the time allowed, and that, as agent_c said, the ship's crew would starve to death long before reaching India. If there hadn't happened to be another continent in between Europe and Asia, Columbus and his men would have died, just like everyone in Europe who had a clue about maths told him he would. Columbus was an incredibly lucky idiot.