Background
The resurgence of trade following the Middle Ages in Europe resulted in a demand for goods from Asia.  Trade routes were established across the Mediterranean and through the Middle East to handle this need.  But, when the expansion of the Ottoman Empire caused disruption along these routes, Europeans were forced to seek alternative ways of importing these goods.  This led to the exploration of water routes to Asia, and eventually the discovery of the Americas by the Europeans.

Science & Technology of Exploration
Advances in technology enabled the Europeans to travel overseas and establish trading empires.

Printing Press: The use of the printing press resulted in many Europeans being exposed to different ideas, such as geography.  This allowed a brave few to venture off into unknown waters without the worry of dropping of the edge of the world.

Gunpowder: Invented by the Chinese, and brought to Europe by Arabic traders in the 1200s, gunpowder was used for military purposes.  European improvements in weapons technology, such as cannons and muskets, allowed them to conquer their trading empires, once they had discovered them.

Naval Technology:  Improvements in mapmaking, or cartography allowed for better navigation.  Also, Europeans learned to use the magnetic compass to determine direction, and the astrolabe to figure out latitude.  The magnetic compass was first used by the Chinese, while the astrolabe was a tool perfected by the Arabs.  The Portuguese built larger ships, called Caravels, to accommodate both the longer voyage, and greater need for space for supplies.  These ships, because of the rigging of their sails and masts, could practically sail into the wind.