What what does the root of american goverment

The American government is modeled heavily after the British government that we broke apart from following the conclusion of the American Revolution. It's ideals of Republicanism are based on the republics of Ancient Greece city states, and the republic of Rome (while it lasted). 

The British Government at the time of our nation's founding, consisted of a king (monarch) and a parliament (legislative). Our government has always consisted of a president (executive), Congress (legislative) and Supreme Court (judicial). The republics of Ancient Greece and Rome were based on elected officials who made political decisions on behalf of the people. Eventually however, these republics slipped into dictatorships when the emperor was given almost total control of all political decisions (similar to Britain before the creation of parliament). 

The American government was the government to incorporate three different branches, each one with their own checks and balances, so that no branch could become too powerful. The president is elected by the American people in accordance with the rules of the electoral college, and U.S. Representatives and Senators are elected by the eligible voters who fall under their territories (county for representative, state for senator). Potential Supreme Court justices are nominated by the president, and voted upon by Congress. The legislator creates bills and votes on them, the president signs bills into laws or vetoes them, and also exercises other executive powers that no other political entity in the United States government possesses. The Supreme Court decides on the Constitutionality of certain laws and court decisions, which can undo standing laws or prevent others from coming into existence. 

Today many governments around the world are modeled after the United States government, which itself was based on the British government at the time, as well as the ancient republics of Rome and Greece.