Short answers, pulled from the story.
A double agent is an employee of a secret intelligence service for one country whose official purpose is to spy on a target organization of another country while simultaneously spying on their own country's organization for that target organization. This role differs from a defector because agents are posted to function for an intelligence service while defectors are not considered agents in the traditional sense.
The threat of execution serves as the primary lever when turning a captured agent working for an intelligence service into someone working for a foreign intelligence service. This coercion transforms individuals like William Sebold, who was coerced by the Abwehr into becoming a spy before exposing the Duquesne Spy Ring to the FBI.
Juan Pujol García known as Garbo became a Spanish double agent in the German spy service and was awarded both an MBE and an Iron Cross for his work between 1940 and 1943. Eddie Chapman called ZigZag is reportedly the only British citizen ever awarded the Iron Cross after infiltrating the German Abwehr while feeding intelligence to MI5.
Aldrich Ames worked for the KGB while employed by the CIA during the Cold War which lasted from 1947 to 1991. Other high-profile cases included Oleg Gordievsky codenamed Sunbeam who was abducted in Moscow in 1985 but escaped to the United Kingdom two months later.
A re-doubled agent gets caught as a double agent and is forced to mislead the foreign intelligence service again after their duplicity has been detected by their original sponsor. A triple agent pretends to be a double agent for one side while remaining truly loyal to another or works for three opposing sides simultaneously.