What where the reasons that people wanted to explore and colonize new lands?

God - Desire to spread their religion (especially Catholicism).  Often used religion as an excuse to justify taking land or enslaving people.

Gold - Desire to get rich.  Sometimes people were actually looking for gold, other times valuable spices, but exploration was a moneymaking venture for both the explorers and their patrons.

Glory- The desire to be famous and bring prestige to their kingdoms.  Monarchs sponsoring voyages often wanted to show the power of their country.  Explorers often wanted to prove they were brave and heroic. 

What new technologies made exploration possible, easier, or safer?

The Astrolabe - Tool developed from Islamic technology.  Helped determine the position of a ship based on the stars or the sun.  Without this tool, exporers would have gotten lost much more often, and probably shipwrecked more often as well.

The Lateen Sail - Triangle shaped sails that allowed a ship to sail into the wind.  Without this ability, ships would have had to wait for long periods of time until the wind was blowing in exactly the right direction.  This would have lengthened voyages, and made getting to some places impossible.

The Compass - Used to determine which cardinal direction the ship was heading.  Used to help make sure the ship stayed on course.

The Caravel - New boat design with more room for food and cargo.  Made longer trips possible as well as more profitable, and thus more likely to be sponsored. 

What are some countries that sponsored some of the earliest overseas voyages?  Why were they willing to do this?

Portugal and Spain were two of the earliest Kingdoms to sponsor overseas exploration.  Because of their position to the far west of Europe, they were at the extreme end of the traditional overland trade routes, but well positioned to attempt journeys around Africa, or across the Atlantic ocean.