The highest voted answer makes sense, but it seems to omit a third possibility, which happens to be the theory I subscribe to.
There are many theories. I like this one the most because it makes sense (to me) and is consistent with how the (real world) universe tries to solve similar paradoxes, e.g. when two objects try to occupy the same space at the same time.
First of all, I think we can agree that if there is no paradox, there is no problem. In other words, everything that's not a paradox can be assumed to work, since it doesn't create any problems. We only need to consider how the universe handles near-paradoxical situations.
The universe will do everything it can (including breaking its own rules) in order to avoid a paradox. However, the universe will always favor using the path of least resistance (= the option which requires the least amount of breaking the rules).
Let's use the grandfather paradox. I go back in time, and try to kill my grandfather (before my father was born) by using a sword. The universe cannot allow that, since it could create a paradox. Therefore, I am unable to lethally wound my grandfather.
- When you try to force two objects to occupy the same space at the same time (by pushing really hard), you simply cannot do this. The objects will break long before you'll be able to have them occupy the same space.
- When you try to bring two very strong magnets (with the same polarity) together, they will not allow you to do that. Whenever you bring them close, the magnetic force tends to make you veer off.
- If you use tremendous force, you can bring the magnets together, as long as this tremendous force is (1) bigger than the magnetic repulsion force, (2) smaller than the force needed for the magnets to break and (3) continuously applied.
Therefore, if I try to stab my grandfather, the universe will make my sword veer off course, missing my grandfather. As much as I try to stab him, the universe will provide a counterforce. Just like how a magnet uses force to repel a magnet of the same polarity, the universe will use a force to repel the action that invariably introduces a paradox.
Suppose I don't intend to lethally wound my grandfather, I only stab him on the hand. Then the universe has several options to deal with this:
- If this wound does not change my grandfather's future (he will still end up having my father, my father will still end up having me), then the universe allows it. There is no paradox.
- If this wound precludes my grandfather from having my father (e.g. his injury makes him use his other hand, therefore not spilling the drink that he's supposed to spill on my grandmother when he is supposed to meet her), then the paradox should be equally disallowed (similar to when I try to lethally wound my grandfather).
Note that I mentioned how the universe favors the path of least resistance. Since the universe is "inventing" a force that would make my sword veer off course, that requires X amount of force. But if the universe has a way to avoid the paradox using less force (less than X), then it will favor this alternate route.
If the force needed to veer the sword off course is greater than the force needed to knock over that drink and spill it on my grandmother (thus introducing my grandparents, even if he wasn't the one who spilled the drink), then the paradox can be resolved this way and the universe will favor knocking over the glass as it is the path of least resistance.
This is consistent with Back To The Future.
- Marty mistakenly intervenes in how his parents fell in love, thus endangering his future. However, the universe has allowed this to happen, as no (unavoidable) paradox has yet been created.
- Marty does start fading away at the end of the movie, because the paradox is getting closer and closer to becoming unavoidable. For example, if Marty took several years to get his parents to fall in love, then they probably would have had Marty at a later stage in life (thus creating a paradox).
- However, so long as Marty's parents are still able to fall in love, on time to give birth to Marty at the right time, the universe has not yet deleted Marty, as he is still possible.
- Marty ends up fixing the problem by getting them to fall in love in a different way. Marty has changed the past, but in a way that the future is unaffected. Changes have been introduced, but no paradox, and therefore the universe does not intervene.
This is the same as the drink that gets spilt on my grandmother. Both the spilt drink and Marty's parents falling in love invariably leads to my grandparents giving birth to my father, and Marty's parents giving birth to Marty.
Minor note: In my example, the glass gets knocked over by the universe. In BTTF, Marty is the driving force behind the correction. However, it's possible to argue that the universe helped Marty along in order to achieve his goal (e.g. making him run faster just so he gets to his destination on time, if being late causes the paradox to become unavoidable). This possibility only applies if helping Marty costs less effort than the universe fixing it by itself.
This is also consistent with Doctor Who.
- In the episode where they are trapped inside a dying TARDIS, the Doctor, Clara and three guests are being chased by monsters (who look like burnt zombies).
- It is revealed that these burnt zombies are actually future Clara and the future guests, who have been burnt by the TARDIS' main reactor.
- Two zombies are attached to eachother. The Doctor realizes that these are the two remaining guests (the third one was already dead).
- The Doctor tells the alive guests to never touch eachother again, because not touching eachother makes it impossible for them to fuse together, thus creating a paradox (the Doctor wants to create a paradox here).
- As long as they never touch, their future selves (the merged zombie) is impossible. And if it is impossible, then the universe should delete that future (similar to how Marty fades once he becomes an impossible future).
- This works for a while, and the future seems to be rewritten. However, the guests forget about not touching eachother, and touch eachother. As if by force, they are immediately joined together and burnt, becoming the burnt zombies.
This force was supplied by the universe. Rather than deleting the future zombies (which takes a lot of effort to rewrite the future), the universe favors bumping the two guests together, as this requires less force and it ensures that the future is actually correct and therefore does not need to be deleted.
Note that Clara has managed to avoid her burnt future. Because the guests have managed to stall the inevitable (touching eachother) long enough, they have changed the future enough to delete burnt Clara (as she is no longer possible), but they did not stall long enough for their own futures to be rewritten.
I can't speak to Looper, as I haven't seen it.
That's proof of how it should work, according to me. We've seen both cases: trying to prevent changes to the future (BTTF) and trying to create changes to the future (Doctor Who).
In both cases, the universe applies the same rules: it maintained logical consistency at all costs, even if that means it has to invent a "phantom force" (which breaks the rules of the universe, specifically the first law of thermodynamics). In order to minimize how much it breaks its own rules, the universe tries to apply the least amount of "phantom force" (= the path of least resistance).