You are completely wrong. No offense intended.
The Commonwealth is the remnants of the British Empire, although for the most part it is composed of completely independent nations.
There are two forms of nations within the Commonwealth Nations - the Realms and the other nations. The Realms are those nations, like the UK, Canada, Australia, etc, that continue to recognize HM as the head of state. At this time there are 16 Realms, although this number can change as realms become republics or if they split up - so, if Scotland gains its independence and retains the monarchy, it will become the 17th Realm.
The other nations are primarily republics with colonial ties to the British Empire, although there are some states that have no colonial ties at all and five who have a different monarch.
The nations, at present 53 in total, are joined together in association because of a common language, history, and culture. They theoretically are all democratic countries with a certain understanding of human rights, and members have been suspended in the part for not living up to an accepted standard. Currently, Fiji is suspended because it's not been democratic in recent years.
The purpose of the Commonwealth is to promote an international cooperation of sorts. According to their website, they work in democracy, economics, education, gender, governance, human rights, law, small states, sport, and youth.