A predestination paradox (also called causal loop, causality loop, and (less frequently) closed loop or closed time loop) is a paradox of time travel that is often used as a convention in science fiction. It exists when a time traveler is caught in a loop of events that "predestines" or "predates" them traveling 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 past in this model, intentionally or not, would only be playing their part in shaping history as we already know it, not changing it (or that the time traveler's personal knowledge of history already includes their future travels to their own experience of the past).
In layman's terms, it means this: the time traveler is in the past, which means they were in the past before. Therefore, their presence is vital to the future, and they do something that causes the future to occur in the same way that their knowledge of the future has already happened. It is very closely related to the ontological paradox and usually occurs at the same time. If Marty had returned to 1985 without noticing any changes in Hill Valley, then the conclusion could be drawn that he had been predestined to be part of its history 30 years earlier. However, at the end of Part I, Marty returns to a Hill Valley different than the one he left, with his family happy and successful, and Biff Tannen answering to George McFly.
The predestination paradox is controversial because it seemingly negates the "common sense" notion that we are responsible for our own destinies and, with adequate knowledge and preparation, we can alter their courses. The predestination paradox says the exact opposite, leaving doubt in believers' minds if they really have free will.
Back to the Future uses several different forms of time travel, and primarily deals with the concept of history being altered, or alternate realities being created. However, several minor details deal with the predestination paradox. For example, in 1955, Marty McFly (Michael J. Fox) discovers that he is the one who inspired Goldie Wilson, his town's African American mayor in 1985, to run for office by accidentally informing Wilson of his future in 1955. Also, by playing "Johnny B. Goode" at the 1955 high school dance, Marty becomes responsible for Chuck Berry's rock and roll composing the very song that Marty would learn to play. His friendship with his future parents led his mother-to-be into thinking that Marty is a nice name, implying that Marty inspired his own naming.
On the other hand, it's clear that before Marty altered history, Goldie Wilson had been inspired by another event to run for Mayor; that Chuck Berry had already written "Johnny B. Goode" after 1955, and that Lorraine had heard the name "Marty" in some other context and thought it was nice. Put another way, those events were destined to happen whether Marty went back into the past or not. Marty's contribution, if any, would have been to reinforce Goldie's ambition, or Lorraine's liking of a particular name.
The closest thing to a predestination paradox might be near the end of Part I, as Marty, having returned to 1985 from 1955, runs across town to the Lone Pine Mall, arriving too late to prevent the Lybians from shooting Doc, and then watching his other self vanish while driving the DeLorean. Since the other Marty hasn't yet departed for 1955, the other Marty hasn't yet altered history by crashing into one of the two twin pines at Old Man Peabody's ranch, and the ripple effect hasn't yet happened, the sign should still say "Twin Pines Mall" as Marty rushes up to it-- unless Marty was predestined to run over the pine tree.
However, it should be noted that the ripple effect isn't instantaneous. It's called the ripple effect because the changes in the time stream are affected like throwing a pebble into a pond. The earliest changes are affected first, and the time traveler (due to being in a temporary time bubble) is affected last. Since Marty hit the pine tree when he first traveled to 1955, the effects of that event have already rippled through time.
If the other Marty lives in a timeline where history had already been altered by the first Marty's presence, then he would encounter the first Marty if he, too, will be arriving at the Twin Pines Ranch on the early morning of November 5, 1955. Since Doc of Timeline 2 (who met the first Marty in 1955) would have had nearly 30 years to contemplate this paradox, then it's possible that in that timeline, Doc would send the other Marty to a date other than during the eight days of November 5 through 12th, 1955. The film leaves the question open, since only the departure from Lone Pine Mall is seen, and the setting of the time machine at the Lone Pine parking lot was taking place while the first Marty was running there from Hill Valley.
However, the Marty that left the Twin Pines Mall is now the same Marty that left the Lone Pine Mall, as he had taken his place. Once the time ripple catches up with a time traveler, the other time traveler is erased from existence. In the new timeline, the Marty that left the Twin Pines Mall overwrote the Marty that left the Lone Pine Mall (possibly when he began to fade from existence at the Enchantment Under the Sea dance in 1955). Doc likely just used his knowledge of the events to make sure that everything played out nearly identically so that Marty would still be sent back to 1955. If he chose another date, Marty would not be able to know when lightning would strike in order to use it to generate the 1.21 gigawatts of power needed to return to the future.
The hint at a predestination paradox is not obvious in the two sequels to the film. The details about 2015, 1985A, and Timeline 8 in 1985 are limited to what can be seen for a few minutes. Arguably, Part III seems to disprove the theory of a predestination paradox somewhat, in that Marty avoids a collision with a Rolls-Royce Silver Cloud III that would have led him to a miserable life in 2015, however Doc tells Marty: "Your future hasn't been written yet. Nobody's has!", possibly indicating that events that have already transpired cannot be largely altered while events that have yet to occur can still be changed from the present. Also of note is that in Part III, Doc ends up saving Clara Clayton from falling to her death in 1885 (seemingly undoing a predestined event), however after he builds a time machine in the past he takes her with him to the future, effectively maintaining her fate of no longer living in the past as he instead moved her with him to the future (although with a 10 year gap from her original time of death).
Though not canon with the Back to the Future trilogy, a video from The Simpsons Ride contains a predestination paradox which involved Professor Frink going into the past to discover how and why The Institute of Future Technology was replaced by Krustyland. He arrived into the past and accidentally ran over Doc's banker Mr. Friedman with the DeLorean. Frink shortly discovered that the accident he caused is responsible for the very thing he came to investigate as Doc found himself having to sell The Institute to Krusty the Clown.