A special election is an election that takes place to elect a candidate to complete the unexpired term of an officeholder who no longer holds the office, due to death, resignation, or disqualification. (In British-speaking countries, the term by-election -- derived from the Old Scandinavian by, meaning town -- is used.)
One of the more interesting special elections in United States history involved the 6th Congressional District of Texas: in 1982 incumbent Phil Gramm was re-elected as a member of the Democrat Party, but due to its increasing liberal economic policies removed him from the House Budget Committee; in response he resigned his office, switched to the Republican Party and ran in the special election, and handily defeated ten other candidates (nine Democrats and one Libertarian) without a runoff.