Answer:
There are a couple of reasons that settlers came to the continent of North America. Here is a list:
Exploration & route to China: Many of the first explorers were trying to find routes to China going west, to establish a trade route with China. They soon found themselves at Present-day Americas, and was able to start settlements while they explore the land. Soon they were followed by real explorers who explored the land more and made contact with the natives in the land.
Conquest: I'm sure you have heard specifically about the Conquistadors, who were Spanish Conquerors who took over most of the Central & South Americas and made them into Spanish colonies. This was also true to the North Americas, as the French, British, and Spanish created settlements and conquered any land to the extent of their control. Many relations were created with the Natives, but usually they broke down and led to the subdue of the Native tribes.
Gold: With the founding of settlements, many of the Spaniards started to search for Gold as well as other natural riches. They generally scoured the area, and would trade for these valuables from natives using beads. Later on they would use natives to help them find these valuables to send back to the mainland.
Liberty for Religion: This is especially true for most British colonists. The British was primarily Protestants, which led to the persecution of Catholics and other groups, such as the Puritans. This led to many groups moving westward, and creating settlements at the coast of the Americas. They usually allowed for the "freedom" of all religions, however this does vary from settlements to settlements.
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