How were government and political structures organized during the Roman Republic?

During the Roman Republic, there were two types of government: the first, exercised by the elite of the Patricians (magistrates, consuls and senators) in a non-egalitarian society, where the rich and politicians were superior to the Roman plebeians. Then, the Roman citizens managed to have the same rights, and all of Italy was organized in Roman colonies, municipalities, and Latin colonies. They were elected two Consuls (supreme commander) to govern the Roman Empire, they only lasted a year and could not be re-elected, and being two consuls, they could avoid the tyranny of one of them, vetoing the other consul their decisions. Little by little the Senate of Rome had both plebeians and patricians, even plebeians came to the office of Consuls.