The first European colonization of the Americas came in the 15-16th centuries CE, after the arrival of Christopher Columbus. Over the next four hundred years, the Spanish Empire explored and colonized a large portion of the Americas, lasting until the revolutionary movements in the 18-19th centuries. The official end of the Spanish Colonial Era was in 1898, when the U.S. won the Spanish-American War. The main reasons for the Spanish colonization were trade routes, royal occupation of territory, native slavery (which was not successful), and the discovery and gathering of gold.
The French colonization came later, in 1524, when King Francis I sent an expedition to seek a route to the Pacific ocean. Their exploration of Newfoundland and the St. Lawrence River led to a number of colonization attempts, each ending in failure. The bulk of French interests lay around the Great Lakes and the upper parts of Canada, where they established forts and trading posts. Their primary goals were to sail through to the Pacific Ocean and find new land and opportunities.
The British were the last to formally colonize the Americas, but they were the deciding force in crucial events during the next four hundred years. Jamestown and tobacco farmer John Rolfe's relations with the Native Americans caused new interest and new growth (incidentally pulling control of the tobacco industry away from the Spanish). Primary reasons for British Colonization included trade routes and resources, military superiority, slave trading, and development of property. The formal British Colonization period ended with the American War of Independence (1775-1783).