A predestination paradox, also called either a causal loop, or a causality loop and (less frequently) either a closed loop or closed time loop, is a paradox of time travel. It exists when a time traveler is caught in a loop of events that "predestines" his or her journey back in time. Because of the possibility of influencing the past while time traveling, one way of explaining why history does not change is by saying that whatever has happened must happen. A time traveler attempting to alter the course of history in this model would only be playing their part in shaping history as we already know it, rather than changing any aspects of the past. This is regardless of the time traveler's intentions or efforts to preserve their personal experience, or knowledge, of events.
The predestination paradox was an integral part of the first movie in the Terminator franchise: The Terminator. There are two main examples where a future time traveler goes back in time and fulfills their role in history (rather than changing it):
- Kyle Reese fathering John - Kyle Reese goes back in time and fathers John Connor, whom he later works under in the Resistance.
- The T-800 endoskeleton being left in the Cyberdyne factory - The T-800 endoskeleton left in the Cyberdyne factory triggers the AI research that eventually results in the creation of Skynet and terminators. (The actual scene showing that the factory in The Terminator is a Cyberdyne facility is in a deleted scene, but these facts are restarted in Terminator 2: Judgment Day).
The predestination paradox is heavily tied to the concept of fate. The photograph of Sarah Connor by the Mexican boy is an example of this: the photo that Kyle had in the future is exactly the same as the one taken of Sarah at the end of The Terminator. This heavily suggests that the events of The Terminator fulfilled the predestination paradox: The Terminator and Kyle Reese traveled back in time to fulfill their roles in history, not to change it.
Terminator 2: Judgment Day
Predestination paradox appears to be non-existent in Terminator 2: Judgment Day. Instead, the theme of the franchise turned away from fate and towards the concept of choice. A key quote from The Terminator was retconned to include the phrase, "There is no fate but what we make for ourselves."
Sarah, John and the Terminator make an effort to prevent Judgment Day by eliminating Miles Bennett Dyson and destroying Cyberdyne headquarters. An alternate ending to Terminator 2 shows that James Cameron considered showing Sarah Connor as a grandmother, proving that the future had been altered. But in the end, he went with the more ambiguous ending that shows up in the theatrical release.
He explained his reasoning in an interview: "But there was a sense that, why tie it up with a bow? If the future is changeable, then the battle is something that has to be fought continuously. And you can't do it with a single stroke. That it's the dualism, the dynamic between good and evil that's eternal." 
Terminator 3: Rise of the Machines
Terminator 3: Rise of the Machines also moves away from the predestination paradox. Instead, the theme is fate: Judgment Day is fated to happen and nothing John does can stop it.
T3 does establish that the timeline has been changed by the events of T2: Judgment Day has been moved back. But fate is eventually inevitable, and Judgment Day can not be prevented - only moved back.
Terminator: The Sarah Connor Chronicles
Terminator: The Sarah Connor Chronicles embraces the mindset of T2. There is no fate but what you make, and it's possible to change the timeline. Predestination paradox has been put aside again, and Sarah and John themselves travel through time in an attempt to change the future.
According to the Voiceovers in Season One, Sarah and company are fighting to stop Skynet from ever being created. If they were to succeed in this mission, then it would stand to reason that John Connor would disappear from existence. If Skynet never existed, then Kyle Reese would never have been sent back in time to father him, Sarah probably would have married someone else, and Derek and Kyle would probably be baseball players.
In a return to the tropes of the franchise, during the events of "Complications" it is revealed that Charles Fischer is yet another character embroiled in the Skynet creation myth and the problems of predestination paradox. Charles Fischer (future) is a member of the Grays faction of Skynet collaborators, instrumental in the instruction and improvement of the T-888 infiltration protocols. He is sent back in time to install future-tech based programming back doors into numerous government and Defense Department computer systems, to ensure the ability of the rising Skynet to seize control. By his future-self actions he directly incriminates his past-self for crimes he had not committed and was responsible for his own incarceration and sentencing to life imprisonment as a domestic terrorist. This is the only reason he was then able to survive Judgment Day and become a captive of and tutor for the machines.
Interestingly enough, as a result of the interrogation techniques used by Derek on the captured Fischer (younger and older), Derek may have further embroiled himself in a paradox resulting in his capture and torture by the elder Fischer in the Alternate future that Jesse comes from, but apparently the TSCC Derek does not.
T2: Infiltrator / Rising Storm / The Future War
The T2: Infiltrator series of novels posits that there is an original timeline, without temporal intervention that saw the rise and fall of Skynet before any time travelers interfered. Once temporal interference created multiple parallel realities, the space-time continuum pushed back and continuously sought to realign with the original unaltered timeline as closely as possible. Here predestination paradox was not enacted by causal loops requiring time travel to become reality, but rather as a force to resolve paradox by re-establishing a specific end state, intervening differences being irrelevant.