What Is The Common Wealth Government?

Commonwealth has a range of meanings. Most basically it is just a name like Republic or State.....
Some American States are still called Commonwealths, like Pennsylvania an Massachusetts. Here in Australia, the official name of our country is Commonwealth of Australia.

It just means “the common good” or the “good of all.”

In Australia, the Commonwealth Government means the national or federal government, as opposed to the State Governments.