Fun With Formality!
America - the United States of America
Great Britain - the United Kingdom of Great Britain and Northern Ireland
Canada - the Dominion of Canada*
Japan - Japan**
China - the People's Republic of China
Germany - the Federal Republic of Germany
(West Germany - the Federal Republic of Germany)
(East Germany - the German Democratic Republic)
France - the French Republic
Israel - the State of Israel
Russia - the Russian Federation
(the Soviet Union - the Union of Soviet Socialist Republics)
South Korea - the Republic of Korea
North Korea - the Democratic People's Republic of Korea
Taiwan - the Republic of China
Australia - the Commonwealth of Australia

*The Canadians insist that they are no longer a Dominion (a self-governing part of the British Empire), but as long as Her Majesty Elizabeth II is also the Queen of Canada, which she is, they remain the Dominion of Canada, est. 1867.
**There is no formal name for Japan. I mean, the dynasty of the Emperor has been unbroken for 2500 years, but the Empire of Japan died in the ashes of Fat Man and Little Boy and the country has never been formally renamed.