Amerigo Vespucci was the first explorer to reach continental lands in the Americas when he discovered the Amazon River and the sites of today's Rio de la Plata and Rio de Janeiro, and he gave South America the name "the New World." In 1507, the region that explorers were learning about was not an extension of the Asian continent and took the name "America" in Vespucci's honor.
Vespucci's first voyage left Spain on May 10, 1497, and according to one of his letters, he may actually have discovered Venezuela before Columbus did. His 1499 voyage took him to the coast of modern Guyana and went on to explore the coast of Brazil, becoming the first European to see the Amazon River. His 1501 voyage was in service of Portugal instead of Spain, and the trip explored the South American coast from Cape Sao Roque down south to Patagonia, discovering what would later be Rio de Janeiro. It was in a letter that Vespucci described South America as "the New World."
In 1538, when cartographer Gerardus Mercator was making his first maps of the New World, he decided to write "America" on both the northern and southern section. While this appeared to give Vespucci credit for lands that others had discovered, the term stuck.