The short answer: The Continental Congress and the states.

George Washington was the first president to be elected according to the terms of the Constitution, which was drafted in 1787, ratified in 1788, and came into effect in 1789. Importantly, the assembly of delegates that drafted the Constitution, the Constitutional Convention, was intended not to create an entirely new national government for the nascent republic (which is what ended up happening), but rather to revise the Constitution's precursor, the Articles of Confederation.

The Articles of Confederation was drafted in 1776 and 1777, shortly after the Thirteen Colonies declared their independence. However, the Articles did not take effect until 1781. The drawbacks to the Articles, among others, included a need for unanimous approval for certain legislation by all 13 colonies and withholding many of the powers from the federal government that modern American government enjoys.

While the Articles were being drafted and ratified, the Continental Congress followed procedures similar to their local colonial legislatures. Many colonies had charters, constitutions, or other governing documents, and many delegates to the Continental Congress had experience in their colonial legislatures.

Before the Articles, the colonies were loosely bound economically by the Continental Association, which was an organized colonial boycott of British goods. This didn't necessarily bind the colonies together, but it created a common economic and foreign policy partnership between them.

While the Continental Congress was the de jure national government, much of its legislation depended on the approval of the individual state. The best example is the vote on the Lee Resolution, which proposed independence. States sent instructions to their delegates influencing their vote on the matter. South Carolina wish to delay the vote for some time, and was somewhat successful in doing so, while New York had yet to receive new instructions from its legislature. Therefore, when the resolution was voted on after several weeks of postponement, New York was the only colony to not vote in favor of independence, though its state legislature eventually passed a resolution supporting independence on July 9.

The title of President did exist before Washington, but it was largely used to refer to an office similar to the modern Speaker of the House or Speaker of Parliament in the UK, which was making sure Congress' sessions went smoothly and orderly. "Presidents" before Washington had no executive power.