What is an election called that is held to fill a public office when the current holder dies or resigns?

A by-election is specifically designed to fill a public office that has become vacant due to the death or resignation of the current officeholder. This type of election is necessary to ensure that the constituents have representation without having to wait for the next scheduled general election. By-elections typically occur in individual constituencies and allow voters to elect a new representative, maintaining the continuity of governance.

In contrast, a general election is a regular election for a government position, often occurring at set intervals, rather than in response to an immediate vacancy. A special election may be used to fill a vacancy as well; however, it is less common and can pertain to specific circumstances beyond just vacancies caused by death or resignation. A referendum is not an election to fill a public office; instead, it is a direct vote in which an entire electorate is invited to vote on a particular proposal, often involving changes in laws or policies.

Thus, the term that accurately describes an election to fill a public office after the incumbent's death or resignation is "by-election."