Going into more detail, I first note that many of the more large-scale societies have entered written history as monarchies, including the first literate ones, Sumeria and Egypt. This raises the interesting question of how far back monarchy can be extrapolated in the absence of written records.
Monarchies have sometimes lasted a very long time, even if they have often been less-than-continuous. Some European royal families, like the Habsburgs and the Hohenzollerns and the Capetians, had lasted nearly a millennium. The Roman Empire's monarchy lasted for almost 1500 years, from Augustus Caesar in 27 BCE to Constantine XI Palaeologus; it was ended by the Ottoman Turks' conquest of Constantinople in 1453. The Chinese monarchy lasted for over 2500 years, from whenever its early history becomes reliable to the abdication of Emperor Puyi in 1911. The Pharaonic Egyptian monarchy lasted for over 3000 years, from Egypt becoming literate to Cleopatra VII Philopator (yes, the famous one); it was ended by the Romans' conquest of Egypt in 30 BCE.
But there were a few notable premodern societies that were republics. Greece emerged from its Dark Ages and became literate again around 730, a set of city-states united by a shared culture. Several of them had rather tumultuous politics, with the old kings being pushed aside or overthrown and going through various combinations of oligarchy, democracy, and strongman rule. Philip of Macedon's conquest of Greece around 350 - 340 BCE was the beginning of the end of this political experimentation. Rome became a republic in 509 BCE with the overthrow of its king, and the Roman Republic grew from a city-state to most of the Mediterranean Basin. But in its last century, it was afflicted by civil war, ending with it becoming the Roman Empire.
Advancing to medieval and early modern Europe, there have been several city-state republics, some of them long-lived. San Marino has been a republic for 700 years, and the Republic of Venice had lasted even longer. The record for a nation larger than a city-state is held by Switzerland, founded 700 years ago.
Switzerland never got many imitators, however, except possibly the Dutch Republic of early modern times. However, the stadholders (leaders) made their position hereditary, making the republic a de facto monarchy. After the defeat of Napoleon, the son of the last stadholder became king.
But the American Revolution was indeed a watershed event in the history of monarchy. The thirteen rebellious colonies of Britain had gotten some practice in republican self-government, and when they broke free, they decided to unite themselves in an overall republic. This republic dwarfed Switzerland, and was about the size of the Roman Republic in its last century. Its first leader, George Washington, refused to crown himself King George I, and he served as President for only two terms, retiring afterward. He refused any titles fancier than "Mister President".
So we can credit George Washington with helping to push monarchy downhill.
Those rebellious colonists had been helped by the French government, on the premise that the enemy of one's enemy is one's friend. But that help proved fatal for the ancien regime, because it gave ideas to revolutionaries. When they took over in the French Revolution, they not only deposed the king, but also guillotined him, along with a lot of other people. France then alternated between monarchy and republicanism a few times before settling on republicanism in the late 19th cy.
In the 19th and early 20th cy. most European nation builders still wanted monarchs for their nations. Holland, Belgium, Norway, Italy, Serbia, Albania, Greece, Romania, Bulgaria. Finland almost got one, but the Great War, as it was called back then, got in the way.
But after independence around 1820, Latin America was mostly republics, courtesy of the US with its Monroe Doctrine of keeping Europeans out. Brazil had a monarchy, but it also became a republic late in the 19th cy.
The Great War (WWI) was another watershed event. That war resulted in the end of the rule of the Hohenzollerns in Germany, the Habsburgs in Austria, the Romanovs in Russia, and the Osmanlis in Ottoman Turkey; monarchies which had existed for some centuries. President Woodrow Wilson demanded, and got, the abdication of the German Kaiser. The Austrian monarch fled, but his position was abolished by the people he left behind. The Romanovs were forced out by civil strife, and then winners eventually executed them. The last Osmanli was forced to abdicate by Turkish nationalists.
These monarchs' successors created republics, which have stayed republics to the present day. The new European nations of Czechoslovakia, Poland, Lithuania, Latvia, and Estonia were all republics. Hungary got a regent, because Hungarians could not agree on which monarch to have -- a Habsburg or a native Hungarian. The new nation of Yugoslavia inherited Serbia's monarchy, however. Portugal had become a republic a bit earlier.
Around the end of WWII, the monarchies of Yugoslavia, Albania, Romania, and Bulgaria were all overthrown by various Communists. Shortly after, the Italian monarchy was voted out of existence, and in 1973, the Greek one shared that fate. The only restored European monarchy has been Spain's, after the death of Francisco Franco in 1975.