Totalitarianism is a political system where the state recognizes no limits to its authority and strives to regulate every aspect of public and private life wherever feasible. Totalitarian regimes stay in political power through an all-encompassing propaganda campaign, which is disseminated through the state-controlled mass media, a single party that is often marked by political repression, personality cultism, control over the economy, regulation and restriction of speech, mass surveillance, and widespread use of terror.
Fascism is a form of totalitarianism. It is based on the idea that the government should control the people. It is the opposite of both democracy and communism because there is no equality or fairness. The masses are controlled via coercion and force, usually by means of a mass-based, militant, ultra-nationalist political party. Also, it is class-based: a small group of people (the ruling class) rules everyone else, who have no basic freedoms or rights: the rulers can arrest, imprison, and kill them as they choose. The vast majority of government-sponsored murder, torture, and imprisonment happen under fascist regimes. Some of the most violent have occurred in Germany, Guatemala, Cambodia, and Chile. A synonym for fascism is dictatorship.
Communism is based on common ownership, mainly concerned with equality and fairness. The power is vested in a group of people who decide the course of action. The group may interfere in the public life of others. It is a classless, egalitarian, stateless society. The government is controlled by one political party and political dissent is not tolerated. It asserts that its citizens are better off without having to choose their representatives, where they live, and what/where they worship. Instead, citizens are assigned jobs and told where they are going to live and what/where they will worship. The government has complete control over the production and distribution of goods and resources, and it is shared in the society equally. Private ownership is not allowed. This helps in preventing any single person or a group of people from raising to a higher position than others or becoming rich, while no one is poor either because the needs of all people are met. The governing body sets the wages of all laborers and regulates the payments, so that no one person earns considerably more or less than another. More work does not necessarily lead to more pay, and this can lead to lowered productivity and advancement. Also, although communism theoretically promotes fairness, the fact that it is based on government control lends itself to abuse and can lead to fascism, and, in fact, often has.
Democracy is similar to communism in that it stands for equality. It is classless (some people consider themselves to be in a higher "class" than others, but that is personal opinion and not part of the democratic ideology). Democracy differs from communism in that everyone is governed by a group of elected officials, who are bound to fulfill the wishes of the society. The masses are free to create their own political parties and compete in elections, which are free of coercion and fair to all contestants. People have the right to choose their representatives and actively participate in the political process. They also have the right to live where they want and worship as they see fit. Goods and resources are owned by private indviduals and corporations. Free enterprise allows people to become rich or poor. Private companies set their own wages and individuals can earn as much as the market allows. More work usually means more pay, and this provides incentive for procuctivity and advancement. This economic system is known as capitalism. It provides more freedom than communism, and of course with more freedom comes more opportunity, but also more risk.
A republic is similar to a democracy in that it uses a representational system where citizens vote to elect politicians to represent their interests and form the government. However, in a pure democracy, the majority can impose its will on the minority, whereas in a republic, a constitution or charter of rights protects certain inalienable rights that cannot be taken away by the government, even if it has been elected by a majority of voters.
Socialism is a mix of democracy and communism. The government operates the system to help all, but there is opportunity for private property and private wealth.
The United States is a democratic republic. However, it has some elements of socialism and communism. There is an overarching government to help everyone, while there is lots of private wealth, so in that sense it is socialist. Welfare, Social Security, unemployment benefits, veterans benefits, public housing, food stamps, Medicare/Medicaid, and anything else that people get for free are communist programs because they are provided by the government to people who don't earn them. (One might surmise that Social Security and veterans benefits were earned by virtue of the fact that people worked for many years or risked their lives, but they already got paid while they were working.) Also, while not fascist itself, the United States has used its military-industrial complex to topple democratically elected foreign governments and install fascist regimes in the name of "fighting communism".
A commonwealth is simply a government based on the common consent of the people, so it's a synonym for democracy and republic. Kentucky, Massachusetts, Pennsylvania, and Virginia call themselves commonwealths.
Imperialism is a policy of extending a country's power and influence through colonization, use of military force, or other means. It cannot be compared to forms of government because it is a foreign policy, whereas government deals with internal control. Any country can be imperialistic regardless of how it runs itself. For example, fascist Germany was imperialistic, as were/are communist Russia and democratic United States.