In 1861, the Confederate States of America began making coins in two very different ways.
The first way was to take over the U.S. Mint facilities at Dahlonega, Charlotte, and New Orleans, confiscate the gold or silver still remaining, and strike coins using left-over federal dies. All coins struck by the Confederacy look exactly like those made by federal employees, so the Confederate States of America still lacked a coin it could call its own.
The second way was for the Confederacy to make coins of its own. Plans were made to have one-cent pieces and half dollars struck at New Orleans.
For the half dollar, the back consisted of a totally new design with a Confederate shield in the center and a Liberty Cap just above. A wreath and the words Confederate States of America encircle the shield and cap. The front of the coin shows the normal Seated Liberty design of the period.
The four known examples are all dated 1861. It's hard to say what one of these would be worth today, because every example is either in an institutional or private collection, with little chance of being offered for sale in the foreseeable future.
We don't know why so few Confederate half dollars were made. The New Orleans mint had plenty of silver on hand and even made over 2 million regular half dollars, which could just as easily have been made into Confederate half dollars.
Later, the Confederate die was used to make 500 restrikes (coins struck again in years later than the date shown on the coin, usually to satisfy collector demand) by grinding off the backs of normal 1861 half dollars and stamping them with the Confederate design.
The Confederate cent has an equally interesting story. The designer was an engraver by the name of Robert Lovett, Jr., who was probably chosen to make cents for the Confederacy because of his previous experience in designing and making tokens. Only a few cents were made before Lovett got cold feet and abandoned the project. Legend has it that Lovett carried one of the Confederate cents with him and accidentally spent it one night in a local tavern. The story leaked out, and a collector of the time hounded Lovett until he eventually sold the dies and the remaining cents. The collector made restrikes in a variety of metals for sale to collectors. Ultimately, the dies ended up in the National Numismatic Collection at the Smithsonian Institution.
So where does that leave the collector who may be interested in these rare coins? The answer, as far as one of the four original half dollars goes, is out in the cold. Any of the 500 restrikes can cost $ 7,000 or more each, depending on condition. Even the souvenir medals in tin (known as Scott restrikes) can run $ 4,000 to $ 10,000 in top condition. The Confederate cent is worth in the $ 30,000 to $ 50,000 range, and a Copper restrike recently brought just over $ 20,000 at auction.