The main motivator for Christopher Columbus was to find a cheap route to the source of spices. Spices were a lucrative trade during Columbus' day. Before the days of refrigeration, food spoiled easily. If you were wealthy enough to afford flavorful spices, you could enjoy better tasting food. Spices, such as cloves, allspice, vanilla, and pepper were well sought after by the elite of Europe. However, they came from far away in South and East Asia. Throughout the middle ages, overland trade was the only way that these spices reached Europe. This trade was mostly dominated by Arab merchants and would pass through many middlemen, meaning that they were very costly by the time they reached European markets. Colombus sought a sea route directly to the source of these spices. He believed that by sailing west he could find a way directly there that would also bypass the routes recently established by the Portuguese who were sailing south and east around Africa.
To a lesser but still notable extent, Columbus hoped to spread Christianity, particularly Catholicism, to the lands he reached. Queen Isabella of Spain, Columbus' sponsor, was an ardent Catholic. She hoped that Columbus would carry the religion with him and convert the people he met in the East to this faith. To that end, Columbus promised the queen that he would be an evangelist of Catholicism in new lands.
We must also consider Columbus' ego as a reason he was looking for a new sea route to Asia. Simply reading his journals shows that he had a large sense of self-aggrandizement. It is clear that he saw himself as destined to be one of history's great figures. He even felt himself to be chosen by God to change the world. This might lend some insight into why he refused to give up when he was repeatedly rejected by potential sponsors until he was able to convince the Spanish monarchs to support his voyage.