Overstrikes are coins that have been struck over a struck coin. Generally speaking, there are two major types of overstrikes. The first type would be a double denomination (a Lincoln Cent struck over a struck Dime). Although these are overstrikes, they are referred to as double denominations and are known on many denominations of coins from many countries.
The other main type of overstrike is a coin which is struck (either deliberately or as a mint error) on a previously struck coin from a different era and country. In ancient times, it was not uncommon to strike coins over previously struck coins with portraits of earlier Kings. In Colonial times, coins circulating were sometimes struck using other coins that were in circulation, regardless of whether they were from the country striking the coin or from a different country.
A rare example of an overstrike is a 1915 Panama Half Commem which was struck in Proof over a cut down St. Gaudens $20 gold piece. Pictured above is an incredible 1970-S Proof Quarter that was overstruck on a Silver Barber Quarter and is one of two known. This mint error was originally discovered in a group of San Francisco Proof Errors that was auctioned by the State of California. There is some detail on both sides showing the design of the Barber Quarter. This is one of the most famous U.S. Proof Major Mint Error ever released from the San Francisco Mint.