There are different ways to categorize different kinds of government. Let us look at two main ways.
First, you can say that governments can be democratic, authoritarian, or totalitarian. These categories differ on the basis of how much power the government has and how much input and freedom it allows its people to have. The US is a democratic country because the government’s powers are limited and the people have a lot of freedom. We could say that a country like China is more authoritarian because the people have little say in government and many freedoms are curtailed. Finally, you have totalitarian states like North Korea in which people have no say in government whatsoever and the state brutally represses people who try to exercise freedoms.
Second, you can say that governments can be federal, confederal, or unitary. A unitary government is one where the national government has all the power and the state (or provincial) governments have little power. Japan is such a country. A confederal government is one in which the states have all the power and the national government has very little power. The UN could be seen as an extreme example of this, and the US had this system for the first few years that it existed. Today, we have a federal system in which a constitution gives some powers to the national government and some to the states.
Hopefully, one of these two answers is what you need.
There are many different types of governments, and each are unique to a country. However, most have one theme in common, and that is that they are elected by the people. This is referred to as democracy. How democracy is achieved however, is a different question.
Generally, governments are made up of a lower house and an upper house. These houses discuss potential laws (bills) separately, and the members of each house vote in favour or against a bill. In the United States, the federal houses are called Congress (lower house) and the Senate (upper house). If a bill passes both houses (gets the majority of votes from each house) it is signed by an executive power (i.e the president in the US) to become a law.