Almost every country in the world is big enough that it has to have a central government and smaller governments. These smaller governments are often called state governments or provincial governments. In the United States, even the smaller governments govern areas that are big enough to make even smaller governments (counties, cities) necessary. Systems that have a central government along with small governments can be classified in three ways.
First, there is a unitary government. This is one in which the central government has all the power and simply gives the smaller governments whatever powers it sees fit. In the US today, state governments are like this. They only give county and city governments whatever rights they want to give.
At the other end of the spectrum is a confederal government. This is one in which the smaller governments have all the power. The central government only has powers that the states choose to give it. This is the system that the US had under its first constitution, the Articles of Confederation. In that system, the central government was very weak and could not even do things like imposing taxes on the people. It had to ask the state governments for money instead.
In between these extremes, we have a federal system. This is the system we have today. In this system, the Constitution gives some powers to the states and some to the national government. Neither the states nor the federal government can simply take powers from the other level of government. The two levels are not necessarily equal, but they each have powers that are reserved for them by the Constitution.