The Age of Exploration refers to the time period between about 1400 and 1650 when Europeans explored and colonized the world outside of Europe. New technology-- taken from the Arabs during the Crusades-- made exploration possible. The search for spices and converts to the Christian faith made exploration desirable.
The Old World generally refers to Europe, Africa, and Asia (the world Continental Europe knew before Columbus). The New World generally describes the Western Hemisphere-- North America, South America, Central America and the Caribbean. These regions were “new” to Europeans.
At the beginning of the 15th century (the 1400s), new technological advances made exploration possible. At the beginning of the 1400s, Europe had just ended the Crusades-- a set of religious wars in the Middle East that lasted 300 years. While conquering Middle Eastern lands, Europeans discovered new technology, such as the compass and triangular sails on boats called caravels, which made exploration possible.
Additionally, the Crusades helped Europeans discover new foods and spices in the Middle East and Asia. What’s so important about spices? Before there was refrigeration, food would spoil easily, making it taste bad. Europeans could use exotic spices like nutmeg, cinnamon, and pepper to flavor their food. Finding spices is what made exploration desirable.
This hunt for spice is what motivated Christopher Columbus to set sail. He was looking for a faster route to India and China so that Spain could trade for spices without having to go through the Arabs and Italians, who controlled the spice trade.
Besides, travelling between Europe and Asia by land was difficult-- there were few roads and there was a lot of warfare. Boat travel was faster, but travelling down around the southern tip of Africa was dangerous (ocean currents frequently blew ships off course). Columbus correctly assumed the Earth was round, but incorrectly assumed that there were no other continents besides Europe, Africa, and Asia. He hoped to sail West from Europe and eventually hit Asia. His theory was correct, but there were two huge continents in the way. (Fun fact-- in each of his three journeys, Columbus never actually made it to the mainland of North or South America. He only explored a few Caribbean islands).
The 15th century was also a time of intellectual curiosity and religious fervor (strong feelings) in Europe. The Renaissance started during this time period, and brought with it a new desire to learn. Many Europeans explored simply because they were curious about the world around them. Furthermore, the fighting against Muslims in the Crusades made many Christian Europeans eager to spread Christianity. Many Europeans explored new lands looking for converts to the Christian faith.
For more information about the Age of Exploration, check out these great eNotes Homework Help answers here and here.