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. The example shown above is an example of a rare overstrike from Mexico. It is a gold 8 Escudos struck over an earlier type and is listed in Krause as rare. The Panama Half and 8 Escudos were intentional overstrikes. There are also examples of unintentional overstrikes which are classified as mint errors.