It is rather hard, reading the definition of 'commonwealth', to see how it differs from 'republic', which comes from the Latin res publica 'public thing' or 'public affair'. Yet it is not the same. The term 'Commonwealth' is only rarely used in official names.
The original Commonwealth in English history is sometimes taken as being the whole period from 1649 to 1660 when there was no king; but it may also be used to mean specifically the periods of Parliamentary rule, 1649-1653 and 1659-1660, as opposed to the near-monarchical rule of the Protectorate (1653-1659), in which first Oliver Cromwell then after his death in 1658 his son Richard Cromwell reigned with the title of Lord Protector.
Four of the US states chose to adopt this term in the following century. Virginia adopted the title on 5 July 1776, Pennsylvania in November 1776, Massachusetts in 17801, and Kentucky when it separated from Viriginia in 17922.
The six colonies of Australia became a federation in 1901, under the title Commonwealth of Australia. But they remained a monarchy under the Queen. Australian institutions tend to use 'Commonwealth' adjectivally (Commonwealth Police, etc.), though 'Federal' may also be encountered. I think this and the Commonwealth of the Bahamas are the only two independent countries that call themselves a commonwealth; the Bahamas is also a monarchy.
The British Empire after its members came to be considered equals from 1930 onward, was first known and is still popularly known as the British Commonwealth, but is correctly known as the Commonwealth of Nations. This phrase is not new; it was used by Edmund Burke in the 1700s.
Apart from the states, two United States territories are called commonwealths; in this case it means they're an associated state (q.v.), autonomous but not fully independent: Puerto Rico and the Northern Mariana Islands.
1. rulers.org/rulu.html#united_states then see individual states for MA, PA, VA
2. http://gov.state.ky.us/symbols.htm for KY