The Commonwealth is a state/community that was started by Cromwell in 1649. It was called “The commonwealth” or “The commonwealth of England”. It becomes the official name of England. “The commonwealth of Nations” is an organisation with the corresponding areas witch has acknowledged the British monarch as the “head” of the Commonwealth.
It is a voluntary corporation of 53 countries, mainly Great Britain and the earlier Britain colonies. But not all of the countries of “The Commonwealth” has the British monarch as there state chief. The ones that have him as there chief are called “The Commonwealth realms”. Mostly of the lands who are members of “The Commonwealth” is republican, but some has there one monarch.
“The Commonwealth of nations” is old British colonies witch have joined this group so that their land is independent, but still in the contact with The British Empire. Lots of countries/old colonies do this because of safety.