Federalism is a political philosophy in which a group of people are bound together, with a governing head. In a federation, the authority is divided between the head (for example the central government of a country) and the political units governed by it (for example the states or provinces of the country).
Currently, several countries in the world have a federal government; examples are United States of America, Canada, Switzerland and Austria. The European Union can be called a sort of federal government as well.
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In the United States Constitution for example, the federal government and state governments both have power. When states and the federal government do not agree on something, the Supreme Court can decide who is right.