Brief Summary
The Set-Up
The colonies declared independence from Britain, officially kicking off the Revolutionary War (though there had already been a few battles). The Second Continental Congress originated the new government of the new United States, written down as the Articles of Confederation.
The Text
As far as governments go, this one was about as simple as butter and toast. The states would send delegates to a national Congress, which would vote on all decisions about war, peace, and federal taxes. Each state got one vote. With thirteen original states, there couldn't be any ties.
Other than that, the states were pretty much on their own. They retained the rights to create their own tax collection methods, organize their own militias, and could put out solo albums at will, sort of like the Wu-Tang Clan.
(In fact, rap collectives are a close modern analogue to what the United States was like under the Articles of Confederation.)
Though it didn't last as long as the Constitution (which is still going strong), the text of the Articles of Confederation included some long lasting ideas. It made titles of nobility illegal (no Sirs, Barons, Earls, or Lords allowed in America), originated the name "U.S.A.," and gave Congress the power to tax…which it still has today.
But the Articles had some major shortcomings that doomed them from the start: they provided no executive branch of government to enforce the laws. They didn't give states with big populations enough say in decision making. And they completely ignored the problem of slavery, which would keep coming up until the Civil War. Before long, it was time to run this thing back.
TL;DR
America was originally organized as a glorified club for the states.