Some would also say 1585, the first attempt made by the English to settle in North America at Roanoke Island off the coast of present day North Carolina. Another attempt was made in 1587 and both of these failed miserably, so as pohnpei states above, 1607 is the most commonly accepted date, as Jamestown was the first permanent settlement.
The colonial period then lasted up until the Lexington and Concord in 1775, and the Declaration of Independence being issued the following year.
I would say that the best date to give for the beginning of the colonial period in what is now the United States would be 1607. This was the year in which the first permanent English settlement was founded in what is now the United States. The name of the settlement was Jamestown and it was founded in what is now the state of Virginia.
People often think the Pilgrims were the first English to settle America. But this is not true since they did not come to North America until 1619.
I believe your question relates to the colonial period in history of the USA. Most historian mark the beginning of colonial period in 1607, when a small band of about 100 English colonists reached the coast near Chesapeake Bay, and founded the first permanent English settlement in North America, which was named Jamestown. Prior to that, in 1585 there was an attempt to settle English People in North America. But this attempt did not succeed.
The Colonial period for the US came officially to an end with the American revolution and the declaration of independence in 1776.