The Age of Discovery lasted from the beginning of the 15th century until the end of the 18th. During this time Europeans travelled far and wide, searching for lands that were until that point unknown to them. Most of these lands were of course already inhabited and as such were not new discoveries for the human race, but as they were unknown to the European explorers who found them, it was still an exciting time where much knowledge was gathered about the world which led to much more accurate and extensive maps than had previously been known.
The reason for the sudden interest in exploring the world was the need to establish new trade routes to the Far East, as the routes that had previously been used were largely blocked by the Ottoman Empire which took control of the area and made it important to find new ways. Some of the most desired goods were spices and silk, but as time progressed more and more emphasis was put on finding silver and gold.
Before the boom in exploration most sailors had largely stayed within sight of land when the travelled, which made it both easier to navigate as there would almost always be landmarks within sight, but also made concerns such as food and water much less important issues; if a ship was running low on supplies, they would simply dock at the next port and take on supplies. Travelling to unknown parts of the world however, the sailors were forced to trust that they would find land before supplies ran out, and that when they did, that they'd be able to find their own supplies as they could not count on finding ports where they arrived.
Some of the countries most active in exploring were the Portuguese, Spanish and Italians, who were soon followed by the French and the British and later, the Dutch.
One of the most well known explorers from this era is probably Christopher Columbus. In 1492 he sailed across the Atlantic Ocean with three ships in the hopes of finding a new and shorter route to India (and the rest of Asia) that did not require sailing all the way south around Africa. He did not find this route however, discovering America instead. He landed on an island east of Cuba, which he called San Salvador. It is unknown however if the present-day San Salvador is in fact the island first discovered by Columbus.
The Age of Discovery continued long after Columbus' first discovery: in fact this was not even the first breakthrough in the Age of Discovery. Many others contributed to it throughout its duration. Even Columbus himself led several other expeditions, but none as well known as the first.