U.S. Commonwealth States
There are 4 so-called "Commonwealth States" among the 50 US states:
Now the question is whether they have another status or not?
No, they don't - it's basically a historical reality and has no political or judicial consequences.
All four states must integrate themselves into the legal system and the political order of the United States - in terms of the United States they are a state under the federal US Constitution, even when they use in their own State Constitution the expression "Commonwealth", see e.g. Massachusetts.
The fact that the four commonwealth states are indeed a state and not a souvereign unit of some
kind is established by their ratification of the Constitution respective the admittance into the Union.
By keeping the expression 'commonwealth' in their name the four states make a symbolic reference to commonwealth units in general,
the Commonwealth of England in particular, and demonstrate their belief in that kind of political unit:
the English Commonwealth was the beginning of the new era after the period of Kings, Queens and kingdoms ruling the country.
Commonwealth in general - the meaning
Commonwealth in terms of one single state or country is defined as a community of people acting in common interest,
resp. the power is held by the people.
The Commonwealth (often referred to as British Commonwealth, what is nowadays no longer correct), describes an actual community
of nations around the world - today more than 50 countries!
They are connected in a social, philosophical, ideological and practical manner and not in a political.
E.g. they are working together on Health- and educational projects.
Even when the modern Commonwealth has its roots in the British Commonwealth, which pooled under this name the countries belonging
to the British Empire, it has nothing to do with a hierarchical system - all participating nations are self-governed,
autonomic and demonstrate the ability to work together without 'borders'.
One mentionable aspect is indeed that The United Kingdom's Queen is anyhow the symbolic head of The Commonwealth of Nations.
Back to the United States: there is still another form of Commonwealth existing: Puerto Rico, which is no US state but a self-governing territory associated with the United States: it's the Commonwealth of Puerto Rico.
And even it's not a state of the US the chief of state is the American president and the head of government is an governor elected by the voters to a four year term.
The people of Puerto Rico are US citizens but don't have a right to vote in the United States.
The four US states, which call themselves Commonwealth, can be referred to as state, too. It is a synonym.