Adequate shipbuilding technology to reach the New World probably existed in the 700s or so, but it wasn't until about 980 that Europeans reached the New World, with the Norse voyages to Greenland and the brief settlement in Newfoundland. Here's a lightly edited version of an older comment of mine on this:

Columbus' discovery of the Americas has very little to do with technology, actually. The Norse settled in Greenland for a couple of centuries and apparently also spent some time in L'Anse aux Meadows, in Newfoundland. They made that crossing in the tenth century, most likely around the year 980, and there's no particular technological reason why they couldn't have kept on exploring down the American coast, then returning up the coast via Greenland and Iceland, and then on to the northern UK and Europe. Their ships used square sails and wind power in part, but were also light enough to row for long distances when the wind failed or wasn't blowing in the right direction.

There are a few theories on why the Greenland Norse settlement was abandoned, and I'm not particularly qualified to comment on those, but you may find some answers in this section of our FAQs. (The Norse were in Greenland for a few centuries. Technically, the Danes are there yet.)

Anyhow, to get to Columbus: The main reason he thought the voyage across the Atlantic would be practicable is that he thought the earth was much smaller in circumference than it is -- I've written about this more here -- and that he had some knowledge of the circular wind patters of the Atlantic, though it's not clear where he got those from.

In Columbus' time, Portuguese explorers had been using carracks and caravels to explore the African coast; those ship types were handier in sailing upwind than earlier types. But early exploration of Africa was limited by the stowage of ships, and how long they could remain at sea without re-provisioning; because sailing upwind took much longer than downwind, the return voyage was a constraint.

In Columbus' first voyage in 1492, he made it across the Atlantic to the Bahamas in about five weeks, following the prevailing westerly winds; he was able to return home by turning north and using the prevailing northeasterly winds and currents off the American coast, which turn east, to return to Europe by a more northerly route. The pattern of currents was known to Portuguese explorers, though the precise source of this is unclear.

I should mention that I had a vague memory of answering a similar question; that answer is here.

Also, I wrote more about Norse longships here.