From March 1, 1781 upon ratification by all 13 states, and until March 4, 1789, the United States operated under the Articles of Confederation.
This vested all federal power in Congress, which was a single unicameral legislature at the time, with no judicial or executive branch. Officially, the 13 states remained individually sovereign and retained all powers, except those explicitly granted to congress, such as the power to treat with foreign nations and declare war. They further agreed to mutual-aid to come to each others defense and citizens of any state were to be granted equivalent citizenship of all states.
Historians now refer to the congress after the signing of the Articles of Confederation as "The Congress of the Confederation", though at the time it was still referred to as "The Continental Congress", the same as the preceding congresses formed during the revolutionary years. In fact, the original membership of the Congress of the Confederation were all the same people already acting as state representatives in the 2nd Continental Congress. It was essentially the same organization, now just organized under different rules. In many ways it carried forward the government previously established during the revolutionary years, which was also limited to a congress, bringing together representatives of the several states.
Congress was to have between 2 and 7 representatives from each state, though each state would receive just 1 vote. In addition to the state representatives, there was a secretary and from among the members would be elected a President of Congress or "President of the United States in Congress Assembled".
From a modern perspective it might be tempting to see the President of Congress as the "head" of the country, but this wouldn't be accurate. This role was not at all like the executive president that came to be under the Constitution. He had essentially no powers beyond any other member of congress, except that he presided over meetings of the congress, acted as moderator during debates and served as chairman of the Committee of the States, an organization consisting of 1 representative from each state that handled national business while the Congress was in recess. In fact, in some ways, being appointed President could decrease the members powers as they could no longer write official congressional documents, but only sign them.
Often, state government positions were more powerful, and many influential men of the time preferred to stay in state positions, and sometimes turned down their election to Congress. Even after elections, it was often difficult to get the states to send enough representatives to form a quorum to do business.
For example, in 1783, then president of the Confederation Congress, Thomas Mifflin, had difficulty getting enough representatives to show up to ratify the 1783 Treaty of Paris to officially end the Revolutionary War. Congress had 3 months to secure ratification, and attempts to secure a quorum through December and January were difficult.
The way ratification worked under the Articles of Confederation was that each state with at least 2 representatives present was given 1 vote in Congress. 9 states were required to pass a resolution under the Articles (technically 2/3s of the states).
Only 7 states had the requisite representatives: Massachusetts, Rhode Island, Delaware, Pennsylvania, Maryland, Virginia and North Carolina. Two more states, New Hampshire and South Carolina, only had 1 representative each and so could not vote. The remaining 4 states, New York, New Jersey, Connecticut and Georgia had sent no representatives.
To make matters worse, the winter of 1783-84 proved to be severe. In New Jersey, storms dumped over 80 inches of snow over the course of the winter, and many roads were impassible even as far south as Annapolis, the capital of Maryland, where Congress was then meeting. As the deadline loomed, Thomas Jefferson proposed asking Great Britain for more time, and sending the treaty, unanimously ratified by 7 states to be offered if they rejected the extension, in hopes they'd accept it.
On January 13, 1784, the two representatives from Connecticut arrived, as well as one representative from New Jersey, bringing the number of voting states to 8. On the 14th, Richard Beresford, a representative for South Carolina was able to arrive, having been sick in bed in Philadelphia but making the effort to travel through the winter storms, and arrived in time to allow South Carolina to be the 9th state for a successful ratification of the treaty. Copies were dispatched to NY by separate couriers, and the French delayed departure of their ship which was due to carry word back to Europe until the arrival of the ratified treaty on January 19th, allowing it to arrive in Paris in the agreed amount of time.
The many problems faced by congress, including its lack of funding, inability to enforce many of its decisions including treaty obligations among others would ultimately lead to the creation of the US Constitution and the establishment of a more powerful central government.
Despite its problems however, Congress did manage to do some important things during this time. They ratified the Treaty of Paris as already mentioned, and passed the Northwest Ordinance of 1787 which established precedent by which the federal government would be sovereign and grow through the admission of new states, rather than allowing existing states to annex new lands further west and also set precedent for the creation of public domain lands.