Anarchy is the absence of government. So it won't be discussed here.
There are many forms (types) of government. Only three major forms will be discussed because all other forms can be distilled to these.
Dictatorships are governments where a ruler and a small clique wield absolute power. Monarchies, socialism, communism, and so-called Islamic republics are all forms of dictatorships. Socialism and communism eventually promise rule by the people, a promise never delivered.
Democracy is a form of government where the supreme power is by the people, periodically renewed. It is the nirvana of governments which, unfortunately, can quickly become mob rule from lack of education and the subversion of the population by the ruling class.
Fortunately for US, our founding fathers realized these pitfalls and created a Constitutional Republic. It is constitutional because the power of the federal government, the States, and the people are defined by the U.S. Constitution. It restricts the power of the federal government and the States. It reserves all other power and rights to the people. It defines a federal government with 3 branches to balance each other.
That is why it is very important for all citizens to know and understand the U.S. Constitution. It defines the laws under which our federal government and the States operate. And our (the people's) power and rights.
It is a republic because the people don't vote directly for the people in the federal government except for our representatives. Our President is elected by the Electoral College and our judges are appointed by the President and confirmed by the Senate.
We Americans have the best system of government and we owe our founding fathers a lot of gratitude.