One could debate at length regarding the beginning of the great European migration to the New World. For our purposes, we will start with the time of Columbus and the claim by both Spain and Portugal upon the new land.
The conflict about ownership was great between those European powers. To settle the conflict, Pope Alexander VI drew an imaginary boundary line (a line of demarcation) from north to south down a map. The line cut through the very eastern tip of Brazil. The two countries agreed that Portugal would own all the land to the east of the line of demarcation, South America. Spain would own all the land to the west of the line of demarcation, North America. See a picture of the line here
The English, on the other hand, were not concerned about treaties or claims of Spain and Portugal. They had interests of their own in the New World. In 1496 King Henry VII issued a patent to John Cabot authorizing him and his sons to "discover and navigate all parts of the eastern, western and northern sea" in the name of the English Crown. Cabot reached the coast of North America in 1497 and then claimed the entire continent for England.
The New World now had two claims of ownership on her, Spain and England.
Then, in 1524 an explorer named Giovanni da Verrazano planted the French flag on North American soil and claimed the entire continent for France. More on Giovanni da Verrazano
In 1585 Sir Walter Raleigh sent an expedition to establish a colony in the New World, and in honor of Queen Elizabeth I (also called the "Virgin Queen") he named all the land VIRGINIA.
Three countries now laid claim to the New World. The conflict would not be settled without several wars and much bloodshed.
But first, there were settlements and colonies to establish. Not an easy task in a new and unknown land!
For one thing, the early settlers had no knowledge of how and where to build suitable housing. Nor did they even know what crops could be grown. Any supplies needed from England would only arrive twice a year on supply ships, IF they survived the Atlantic crossing.
For the first hundred years, all new settlements were located on the eastern seacoast, or a short way up some of the navigable rivers. There had to be easy access to the ocean in order to receive supplies. During this time, the settlers became acclimated to the new land and increased their self-sufficiency.
By 1700 America was pretty well established and more and more she was being seen as a "Land of Opportunity". However, the frontier had only advanced westward about 100 miles because clearing the land was such a monumental task. Immigrants were now arriving in increasing numbers. But all the available land was already claimed and under cultivation. Soon, the crowding would neccessitate a western expansion and even more unknowns.
1) Laurence Bergreen, Over the Edge of the World: Magellan's Terrifying Circumnavigation of the Globe (HarperCollins Publishers, 2003)
2) Jack Greene, Pursuits of Happiness: The Social Development of Early Modern British Colonies and the Formation of American Culture (Chapel Hill, 1988)
3) Samuel Eliot Morison: The European Discovery of America. The Northern Voyages A.D. 500-1600. New York: Oxford University Press (1971)