Short answers, pulled from the story.
A double agent is an employee of a secret intelligence service whose official role is to spy on a target organization of another country, but who is actually spying on their own country's organization for that target. The role requires sustained deception of at least one controlling organization.
Execution is the most common method used to turn a captured spy into a double agent. When an intelligence service catches a foreign agent, the threat of death is frequently used to compel the agent to switch sides. Infiltration by a target organization that plants agents inside the controlling one is another route.
A re-doubled agent is one whose duplicity as a double agent has been detected by the original sponsor, who then pressures them to reverse sides again. F.M. Begoum defined the category as someone persuaded to reverse affections a second time after being caught doubling.
A triple agent pretends to be a double agent for one side while genuinely functioning as a double agent for the other, and has typically maintained loyalty to their original side throughout. A re-doubled agent, by contrast, switches sides a second time because they were compromised, not out of sustained original loyalty.
The Double-Cross System was a British wartime programme that ran double agents against German intelligence during World War II. Known agents included Mathilde Carre (La Chatte), Roman Czerniawski (Brutus), Eddie Chapman (ZigZag), Juan Pujol Garcia (Garbo), Walter Dicketts (Celery), and the Norwegian pair Moe and Glad (Mutt and Jeff).
Juan Pujol Garcia, code-named Garbo, was a Spanish double agent who worked for British intelligence inside German spy services during World War II. He is notable for being awarded both an MBE from Britain and an Iron Cross from Germany, the dual honours reflecting how completely he deceived both sides.