An heir is a person in line to receive an inheritance. It is most often used in relation to the person who is next-in-line to become king.
The heir-apparent refers to the first-in-line for the throne in cases where nobody else may be born to supersede that claim. In cases where the person currently next-in-line may yet be superseded, such as when the current monarch has not yet had children, the heir is referred to as the heir presumptive.
In France the heir to the French crown was known as the dauphin. Since 1301 the heir-apparent to the English crown has been made the Prince of Wales.
Heirs in the SCA
In the SCA, the winners of the Crown Tournament are regarded as heirs to the throne and are titled Crown Prince and Crown Princess until they are crowned.
Likewise the winners of a coronet tournament are the heirs to the Principality throne and are sometimes referred to as Tanist and Tanista.
Baronies also may have heirs, and although there is no formal and universal term for them, in the East they are often referred to as "Their Nextellencies."