To govern is to rule, regulate, control, dictate, or command. Noah Webster formerly included a single distinction: “either by established law or by arbitrary will.” Government is the person or group of persons who govern. either by established law or by arbitrary will.
Once reduced to their basic character, there are just four ways to govern. One is by the mass of the people (democracy, or majority rule), by the will of the minority (oligarchy), by sovereign will (monarchy, communism, or dictatorship), or by God’s will (theocracy.) Some would add rule by nobody, but that is not government. Anarchy is the absence of government.
Within those four ways of governing, there are many forms of government and structures, combinations and rules for establishing, implementing, and enforcing law. For instance, a monarchy may depend more or less on an oligarchy or democracy, a democracy may appoint an oligarchy or dictator, and so on.
(Some will strenuously object that communism is not “sovereign will, but the true will of the majority!” That would be true if all the people agree with the dictates of the sovereign leader, or “First Citizen.” At best, the sovereign leader will accept the will of the majority, but without provision for dissent.)
What, then, about a republic, or “the rule of law,” as we once had in the USA? A republic is not a way of governing. A republic is a form. It informs and restrains the way of God, of the monarch, of the minority, or of the majority in the human system. In other words, a republic governs (rules, regulates, controls, dictates, and commands) the government by any means. It compels the government to do some things, and prevents the government from doing some others.
A republic (rule according to the limits of the written law) is supposedly the US form of government. Other forms all include rule by will or decree.