The commonwealth is, literally, things that belong to the public. There are specific commonwealths, such as the Commonwealth of Virginia, for whom the commonwealth is a naming convention that doesn't differ from the definition of a 'state' by U.S. laws, or the Commonwealth of Nations, which identify a group of countries with a common connection to British royalty but little else in law. A commonwealth is a way to describe an association of individuals who agree on how to separate private ownership from what belongs to the larger group to benefit the common good.
Because a commonwealth is often used to describe a sovereign group or entity, there's a tendancy to assign the term its own separate club, as though a "commonwealth" has a fancier set of rules. This is untrue; the character of a commonwealth is established by its components, and not by the title itself.
As a pormanteau, the "common wealth" would describe public places and property, and is an accurate description of a commonwealth's purpose. The words themselves are the English translation of the Latin term Res Publica, the source of the word "Republic". "Res" is a general word meaning "things", as in a tangible or real object; "Publica" refers to the public. Res Publica therefore literally describes the "public's things". This is translated more directly in French, which calls republican ideals the "chose public", with "chose" being another general word meaning "thing" or "stuff". Owning "stuff", of course, means an asset, which brings us back to describing the "common wealth", those things which are owned communally for the benefit of all.
Now, both "res publica" and "chose public" are generally translated as "public affairs", because government owns more than purely physical things, such as treaties and armies and laws and debts. This definition applies best to modern interpretations of a commonwealth's purpose and scope. It is a term interchangeable for most states and nations for whom the public is a focus of governmental interests.
What further makes a "commonwealth" a far more general term is that it does not inherently identify a form of government. Governments established for the "public good" tend to be lumped together as exclusive to democracies, but utopian communism easily fits within the description of a commonwealth in their communal ownership of a significant portion of property for the benefit of, and at the behest of, the public good. Also, in the case of the Commonwealth of Nations, that commonwealth descibed the organization of nations, and not necessarily the government structure of any of the member nations. The UN, NATO, Warsaw Pact, EU, and other international associations fall under the broad umbrella of a commonwealth of sovereign countries. The United States of America, being a republic composed of democratic states, could be considered a commonwealth of commonwealths.