Italian dude who couldn't convince Italy to pay for his Asia trade route exploration adventures, but convinced Spain to sponsor him to find a Western trade route to Asia and winds up in America  (which he thinks is Asia!).

Another Italian dude sponsored by Spain. Vespucci was the first European to figure out that America wasn't actually Asia. He realized that Brazil was, in fact, not Asia and that the land he was exploring was actually a completely different continent. The Americas are named after him.

Spanish conquistador and first known European to reach the mainland of present day United States. He's also the first governor of Puerto Rico...and he's the guy who named Florida.

Spanish conquistador and head of the first South American settlement. He's also the first European to discover the Pacific Ocean

Portugese dude who was sponsored by Spain to find a Western trade route to the Spice Islands and ended up leading the first circumnavigation around the world. (He actually died before the trip was over, but his crew made it back to Spain...so it counts.) He's considered the first European to cross the Pacific Ocean.

Spanish conquistador who was obsessed with quick wealth and power. He's a pretty awful guy. He is responsible for the arrest that led to the death of Vasco de Balboa and the end of the Inca civilization in South America. 

French explorer sent by France to explore the New World.  He's the guy who named Canada and claimed it for France. He was the first European to find the St. Lawrence River.

British sea captain, privateer, and slave trader. He's the first European to circumnavigate the world and stay alive to see the whole thing.  Also, he's the guy who led the victorious attack against the seemingly invincible Spanish Armada (a.k.a. the Spanish Navy fleet). He basically ended Spanish dominance in the New World.

British dude who attempted to establish the first permanent English settlement in the new world: the Roanoke Colony. He's the guy who wrote a book that made the treasure of El Dorado a legend. He's also the guy who introduced potatoes and tobacco to England.

British sea captain who brought people over to found Jamestown settlement - the first successful permanent English settlement in the new world