The Commonwealth of Nations is often referred to as the British Commonwealth or just the Commonwealth.  There are 56 member states, most of which are former members of what was known as "the British Empire". 

They are connected by language, history and culture.  Members include the United Kingdom, the former Dominions of Canada, South Africa, India, Australia and New Zealand.  They also include British Overseas Territories and Crown Dependencies.  It is headed by Charles III.  The Commonwealth was formalized by the Statute of Westminster in 1931.  There are 15 Commonwealth realms, including Canada.  There are 36 republics.  There are 5 other members that have different monarchs (Brunei, Eswatini, Lesotho and Tonga. {Yes, we have heard of Lesotho}.    

Commonwealth citizenship affords some benefits in some member countries - for instance in the United Kingdom, one can volunteer for military service in the British Armed Forces if one is a citizen of a Commonwealth member country (but it does not confer UK nationality).  The Commonwealth Charter defines shared values of democracy, human rights and the rule of law.  This is promoted in the quadrennial (i.e. every four years) Commonwealth Games.  In 2019 the Commonwealth had a combined GDP of over $9 trillion, with 78% of that from the four largest economies of the Commonwealth (India, United Kingdom, Canada and Australia).  The USA had a GDP in 2018 of $20 trillion. The EU had a GDP of 19 trillion.  The Commonwealth has a combined population of 2.7 billion - far in excess of that of the USA or of the EU.