Which countries did most of the exploring?

Of course, this question depends on the time period and your definition of "explore." Remember, when Europeans traveled outside of Europe to Asia, Africa, and the Americas, they found people already living there, so technically the land was already explored. There is a growing movement in historiography which focuses on the indigenous tribes of North America and their historical contributions.
That said, one can argue that the Portuguese and the Spanish made the greatest contributions to the West's discovery and ultimate mastery of the world. Both countries often used explorers from the Italian city states; for example, Columbus was from Genoa. In 1494, Spain and Portugal signed the Treaty of Tordesillas, thus dividing the world outside of Europe into Spanish and Portuguese territories. In time, English and French explorers would also make vital contributions to exploration, but they used the nautical maps of these Iberian explorers as a guide.