List the five main european countries active in exploration.

Those would be Portugal, Spain, The Netherlands, France and Great Britain.
Portugal was the nation that started the Age of Exploration, and it is famous for its travels around Africa. Like all nations in the Age of Exploration its goal was finding reaches. In the New World they colonized Brazil.
Spain was a major player in the New World and its Conquistadors allowed it to colonize the Maya and the Aztec and conquer their civilizations. They settled in the Latin America mostly.
France was a nation that fought mostly for Africa but they had colonies in the New World also. They were a big fur traders and explorers of the North America.
Great Britain as we know held most of the North America until the American Revolution and was  a dominant power there. The stories of the explorers and settlers are well known across the world.
The Netherlands are maybe the most well known for their purchase of the Island of Manhattan from the Native Americans. Other than that they did not have many major colonies in the New World.