A federation, in Webster1913's last sense, is a state composed of multiple mini-states that maintain a limited degree of
autonomy. Federations are most often employed when
governments need to create a unified rule among a diverse population.
Two basic models of federation exist. The USA's model, which is more common in the Western Hemisphere, emphasizes the independence of the constituent states. Germany's model, which is more common in Europe, emphasizes the cooperation of regional and federal government.
The mere existence of regional governments does not mean that federalism is being employed in a country. France and Japan have regional governments (departments and prefectures respectively), but these only serve to enforce national laws that are made in Paris and Tokyo. In a true federation, the regional governments can set their own laws: in the United States, for instance, each has a different speed limit, age of consent, income tax, capital punishment, and education structure.
Is a federation (see individual nodes for juicy details):
United States of America - 46 states, 8 commonwealths, and a district
Canada - 10 provinces
Mexico - 31 states
Germany - 16 Länder
Austria - 9 Bundesländer
Belgium - 3 regions
Switzerland - 26 cantons
Russia - 50 oblasts, 21 republics, 10 okrugs, and 6 krays
Yugoslavia - 2 republics
Bosnia and Herzegovina - 2 divisions
India - 25 states
Pakistan - 4 provinces
Nigeria - 36 states
Comoros - 3 islands
Venezuela - 23 states
Brazil - 26 states
Ethiopia - 9 states
Malaysia - 13 states
United Arab Emirates - 7 emirates
Australia - 6 states
Arguably on their way to becoming a federation:
United Kingdom - devolving power to Scotland, Wales, and Northern Ireland through regional parliaments
European Union - seeking to create a united economic and political front for otherwise independent nations
Iraq - one proposed model of government
Could have been a federation:
Commonwealth of Independent States (but nobody cared)
Was a federation:
Confederate States of America
East African Federation
Federal Republic of Central America
Granadine Confederation