In brief, the office of the president did not exist until US Constitution came into force in 1789.

In 1781, the 13 states ratified the Articles of Confederation. The Articles of Confederation served a constitutional function, but unlike the later/present US Constitution it reserved much more power for the states and created a much weaker federal government. The Articles of Confederation did not establish a national executive branch, with no office of the presidency or position equivalent to it; while there were presidents of Congress, the job was quite different and much less important in terms of the powers granted to the office.

For a few years, the Articles of Confederation served as the United States's unifying constitutional document. But during and after the Revolutionary War, the weakness of the federal government under the Articles created a variety of problems. Crucially, the federal government under the Articles struggled to levy taxes, regulate commerce, suppress rebellion, etc. In mid-1787, 12 of the 13 states sent delegates to a convention in Philadelphia to propose changes to the Articles of Confederation. Instead, however, these delegates proposed scrapping the Articles and drafted a new US Constitution. This new Constitution included the office of the presidency, who would serve as chief officer of their proposed executive branch of the federal government.

Once the delegates proposed this new Constitution, it went to the states for ratification--requiring a minimum of 9/13 to take effect. This took some time, as the new Constitution was controversial and met significant opposition in many states from Anti-Federalists. It took until June 1788 to reach the required 9 states, and another month to get Virginia and New York to sign on (crucial to the success of the Constitution, as they were the most populous and powerful states at the time). The first presidential election was held the following winter; Washington won the election (with unanimous support from the Electoral College) and was inaugurated as president the following Spring.

For an more detailed survey of this history, check out Robert Middlekauff's The Glorious Cause: The American Revolution, 1763-1789. It's a bit of a dated work in some ways (originally published about 35 years ago) but the last few chapters will serve as a useful introduction to the early history of the Constitution and the creation of the office of the president.