All the factors already mentioned were obviously and undeniably important. Yet human beings throughout history have been wanderers, moving first out of Africa and then into most known corners of the old and new worlds. The motives that prompted the earliest humans to cover the face of the earth were probably not much different from the motives already mentioned in connection with European overseas expansion. Likewise, the motives that prompted both the earliest humans and Renaissance Europeans to explore and settle the earth will probably someday be responsible for exploration and settlement on other planets. Humans have always tended to be, and are always likely to remain, wanderers.
Another less important yet still humanly vital reason for European overseas expansion was the thrill of exploration and adventure. Remember, Columbus had reminded those newly emergent from the Dark Ages that the world is a round globe (physics has modified the shape, of course) and not a flat plane from which one could be lost forever should one venture too far (an idea C.S. Lewis makes neat use of in Prince Caspian). There was a whole new world and a whole new concept to discover.
One of the contributing factors for English overseas expansion that has not been mentioned is the fact that the small island of England wanted some place to send its many criminals (since being in debt and so many other things were crimes). Australia (New South Wales) was originally a penal colony, as was a part of the state of Georgia in America.
Imperial competition was a very important factor. Competition in terms of access to resources, economic competition, a struggle for the control of sea lanes and trade, and religious competition. Many of these European empires were in direct conflict with each other on a regular basis and flt as though the global race to expand and acquire colonies was inevitable, and that if they did not participate in it, their empires would eventually be defeated.
Wealth, greed, prestige, power, fear of being left behind. That pretty much sums it up. Of course, it was the belief of cultural supremacy that likewise "supported" European expansionism and the way in which taking Christianity to the uncivilised masses of the world played an important part in this that was often used to "screen" such sordid and base motives.
I agree with the other posters that there were number of tangible goals including money, spices, and a desire to expand influence. I also think that Europeans had a sense of entitlement and arrogance that I think is innate to their culture. Like many cultures they believed that they were special, chosen by their god or gods as gifted in some way. This sense of uniqueness I think is common among all cultures, but the political and geographic makeup of Europe allowed this idea to grow, coupled with the rise of nationalism and improvements in technology, compelled nations like Spain and Portugal to explore the world claiming all that they found.
Greed. Everything that led the Eupopeans overseas had to do with greed. They wanted gold, spices, more land and basically anything that could make them money. They wanted all of these things faster. So they explored and found new routes, and new goods to trade. They happened to hit the jackpot and found more countries to colonize along the way!
God, Gold and Glory was indeed the avowed aim of the Spanish Conquistadores; yet they were only one of several groups of Europeans who engaged in European expansion. A number of other factors were at work. A large factor was the capture of Constantinople by the Turks in 1453. Europeans had previously purchased spices and Oriental goods there, but had no desire to trade with Islamic "infidels." Many nations wished to find their own route to the East and avoid Constantinople. The first to successfully do so were the Portuguese, when Vasco da Gama reached India. He informed the Indian emir that he came for "Christians and Spices."
The European Renaissance was also a major factor. The increased emphasis on experimentation, knowledge and learning led to superior methods of navigation and ship building, many of which were borrowed from the Muslims. This included use of the Astrolabe, the lateen sail, and the Compass. This made sailing over long distances more feasible.
The consolidation of nation states in Europe meant that monarchs now had the means to fund overseas exploration. Also some exploits were privately funded (Christopher Columbus privately funded two of his ships, only the Sana Maria was funded by Ferdinand and Isabella) most were pursuant to royal charters. John Cabot (Whose real name was Giovanni Caboti) sailed for England, even though he was Spanish. Henry Hudson sailed for the Dutch even though he was British.
Exploration along the coast of Africa was spurred by the search for a mythical Priest King named Priester John. When Prince Henry the Navigator sent expeditions down the Atlantic Coast of Africa, they carried a letter from the Pope addressed to Priester John and written in Latin asking for his help in expelling the Muslims from the Holy Lands.
Finally, Marco Polo's book, A Map of the World, inspired great interest in exploration and the search for riches. Although many portions of his book are suspect, and there is even substantial doubt as to whether he actually existed, his book did create tremendous interest in Asia. Christopher Columbus owned a personal copy of the book with notations in the margins in Latin.
The classic way to enumerate these factors is to say that the expansion was precipitated by "God, gold, and glory."
During the Age of Exploration, Europeans were motivated by a variety of factors. To some extent, countries and individuals wanted to spread the Christian religion across the world. One factor, then, would have been the belief that Christianity was the one true religion and that non-Christians would benefit from being converted. Expansion was also motivated by the desire for national and personal gain. People and nations wanted to become rich and to gain prestige. Another factor, then, would be the desire to gain power.
Of course, all of these desires would have been impossible to fulfill had it not been for the Europeans' technological advances. One might say, then, that the development of technology was the major factor that precipitated this expansion.