How did the United States gain its independence from Britain?

The United States gained its independence from Britain by winning the Revolutionary War (1775–1783). This is in contrast to Canada, which gained its independence from England in a gradual and peaceful way.
The American colonies had gotten along with the mother country fairly well for most of the relationship. Colonists settled the Eastern seaboard of the North American continent in the early seventeenth century. Because of mercantilism, the economic relationship favored London's interests, but the relationship between the colonists and the king was relatively harmonious. Even after the first shots were fired in 1775, many colonists—perhaps as many as one-third—remained loyal to England. Many of these so-called Loyalists eventually settled in Canada.
The Americans—aided by France and other foreign powers—eventually won the war. England had to give up her thirteen American colonies, but she remained an important colonial and world power.