British North America was the British-owned part of the continent of North America which constitutes today's America and Canada. This term was mostly used before the American Revolution.
In 1759, after winning the Battle of the Plains of Abraham, Britain took control of the colony of New France. After this, British North America also included New France.
The term "British North America" was used after the 1783 Treaty of Paris. This ended the American Revolutionary War, and said Great Britain's Thirteen Colonies were independent.
That formed the United States of America. The terms British America and British North America continued to be used for Britain's remaining territories in North America. The term "British North America" came to be used more consistently for the provinces that would eventually form the Dominion of Canada.
After the American Revolution, some people still used the term to refer to what is now the country of Canada. In fact, the Canadian Constitution was first called the "British North America Act 1867".
Also note that the areas beyond the first American states were not at first part of the United States. For example, areas near the Mississippi were not part of the original United States. They kept their old allegiances to England or France. And, of course, what are now the western states were largely Spanish-speaking until the later United States took them by force. Lastly, the division of North America into a lower part called the United States, and a northern part, called Canada, lay in the future.