So generally there are three ways people interact with time travel, all based on Paradoxes that spring froth from a Paradox and the results of those travel.
The Grandfather Paradox
The Grandfather Paradox gets its name from the idea that if you, a time traveler, were able to prevent your biological grandfather from ever meeting your biological grandmother, this would cause the universe to collapse, since it creates the problem that you never existed, thus could never time travel to prevent your grandparents from meeting, thus your grandparents meet, thus you were born and could prevent them from meeting. In it's worst implementation, the universe enters of infinite loop where the entire universe stalls on this point because it never resolves. This does not mean that the past cannot be changed but that the time travel must be careful not to change the past in a way that removes their initial cause for the time time travel. Consider "Back To the Future" where Marty McFly accidently prevents his mother from initiating a romantic interest and thus Marty runs the risk of being erased from existence. However, he is able to prevent Doc Brown's death by alerting him to the nature of his own death, and some of the circumstances of his parent's initial romance, both of which do result in changes in his own personal history, albeit for the better. Typically the central conflict in stories with the Grandfather Paradox is preventing the paradox from getting introduced in such a way that the universe does not reach the paradox. In best case scenarios, the universe will resolve in ways that will prevent this from happening, where we get other Paradoxes:
The Butterfly Effect/For Want of a Nail/Multiverse
This is a resolution of the Grandfather Paradox where the universe does not reject your attempts to change the past or shorts out if you do and you are returned to the present with the changes in place. This gets many different names, but the result is the same... the present is now changed to a point that is usually worse compared to the present timeline (Not always. "Back to the Future" resolves in a positive changes (for the Protaganist, at least. The Negative Changes are what motivate Biff Tannen in the sequel)). The Butterfly Effect is named for a scenario that posits that every interaction someone has in the past will have huge changes on the future.
In effect, the flap of a single butterfly's wings, the change in the atmosphere can result in a massive Hurricane elsewhere in the world. This is demonstrated in the film with the same name, where the protaganist tries to use time travel to make his friends lives better, but ends up making them worse (ironically, the best result makes everyone happy... but leaves him exclusively in a much more miserable life). Not only that, but since he can only return to certain points, the more he meddles, the less he has the ability to return. "It's a Wonderful Life" a similar concept is employed (although time travel is less explained) with the Angel Clarance granting a suicidal George Bailey a chance to see what a difference he made by his existance up until that point by showing him a world where George was never born. Without his sacrafices, the town he lived his whole life is a crime ridden cesspool, but what's worse is that because he never saved his War Hero Brother while they were children, countless people died because his brother wasn't there to save them. "For Want of a Nail" comes from a proverb/poem that posits that because someone could not find a nail in his workshop, the entire kingdom fell in a war. The moral being that something as cheap as a nail is still important in the long run. This can be seen in real life all the time when looking into reports on the causes to many man made disasters resulted from a tiny piece of infrastructure that failed to prevent a disasters chain reaction. Many stories featuring this time kind of time travel will be set by showing the bad future first and sending the protagonist back in time to set right what went wrong. In it's ultimate expression, the "bad future" and the "good future" exist at the same time in parallel universe, along with universes that exist for every other outcome to every action in all of time, creating an infinite "multiverse" of possible timelines, some with barely noticeable differences and some with staggering changes. Often times the point of stories involving the Mulitverse is not so much changing the past, but exploring the possibilities that differ from the story's original time line. Marvel Comic's "What If" Line and DC's counterpart series "Elseworld's" will typically take a moment in their character's origin story and propose a change and then show how that change affects their world. Marvel's What If tend to be more grounded on key events and famous story lines (What if Spider-Man saved Gwen Stacy?) while DC's tend to be more shifting settings (Superman: Red Sun posits a World where Superman lands in the Soviet Union and becomes a champion of Truth, Justice, and the Marxist Way! while "Gotham by Gaslight" takes the Batman mythos and puts it in a late 19th century setting.).
The Predestination Paradox
The Predestination Paradox occurs in such a way prevent the Butterfly Effect and Grandfather Paradox from ever happening, but preserving Time Travel as a possibility in the story. In effect, Time Travel can happen, but the events of time travel are factored into the current history. In real life, there is debate about who actually fired "The Shot Heard Around the World", the first shot in Battle of Lexington that resulted in the stand off between Colonial Minute Men and British Troops escalating into a shoot out that became the first battle of the American Revolutionary War. In a story about time travel, a historian would travel back to the day of the battle to learn the truth... only to discover that through accident or through his own actions, he was the cause of the "Shot". That is, in a Predestination Paradox setting, history isn't changed by the actions of the time travel, but rather it happened as it did even with the time traveler present or specifically because of the time traveler's presence. In this setting, all of history will occur no matter how it happened and any change to the time line would already be in effect prior to the time traveler's departure from the present. If the future is the destination, the events of the future will come to be in a self-fulfilling prophecy... any attempts to prevent a future event will contribute to that future event coming to pass. In some cases, this can be so enforced, that all actions to prevent the time traveler's goal seem so unlikely that it is to suggest that the universe is actively preventing you from getting close to your goal... and the more likely you are to making the change, the more bizarre the preventative measure. In other cases, the events that involve the time traveler fill the gaps or legends of the known story. Another interesting phenomena is that these events must happen to the exact letter of the prediction... but so long as the condition is filled, it might not be the doom that it was initially thought to be. Because Gargoyles made this a hard rule with it's time travel stories, there are alot of examples. In one episode, Goliath arrives in London to see a monument to himself and another Gargoyle he's never met, honoring their sacrafice during the London Airraids during WWII (Goliath is over a millenium old, but he was frozen in stone until 1994... there was no way for him to do much during WWII). He tracks down the donors who funded the monument and they are shocked and offended that he would show up without their friend Griff (the unknown Gargoyle). Goliath says he doesn't know Griff, to which the donors explain that Goliath showed up one night during the war, agreed to help Griff and promised the two strangers that he would keep Griff safe. The pair leave and that was the last night either of the two donors saw Goliath or Griff. This not being Goliath's first time dealing with time travel (and him having the components to do it) quickly time travels to the night in question to find out what actually happens. The events up to their departure transpire as described, and once up in the fight Goliath and Griff have some initial sucess against the German Pilots... but soon, things start to go south... not catastrophically so, but the pair are having more and more near fatal close calls... with Griff being the unlucky of the two... It's here where Goliaith realizes that they were fated never to be seen by Griff's friends after they left... and the friends were kept in the dark about Goliath's time traveling... and realizes that the donors only assumed that Goliath and Griff died... and even the universe seemed content to kill Griff. Goliath realized that he could still bring Griff home safe... just not that night... or any night before he encounter the mystery... and uses his time travel spell to take Griff to London of the mid-1990s moments after he departed WWII London, thus keeping his promise while not changing the past at all. No one found Griff the next day because he had bypassed that day... and nothing was preventing Griff from existing in the future. Another example of this occurs in Doctor Who with the relationship of The Doctor and a woman named River Song. When the Doctor first meets her, she knows things about him that he has at this point never told anyone and was given a gift of a sonic screwdriver (The Doctor's signature weapon/tool), which again, he's never given anyone else... let alone someone he's never met before. River explains that the Doctor was vitally important to who she is but refuses to tell him who she is... according to her, at some point in one of their prior encounters the Doctor explained to her that the doctor's first meeting was on the day she died and she can't tell him everything about her because they are quite literally "spoilers". As it turns out, River is a time traveler like the Doctor, and because of this, every time they meet, each is in a different point in their life such that each is essentially meeting a progressively younger version of each other. As such, as the arch progresses, the Doctor and by extension the audience learn more about her while her character knows less about the Doctor each time, such that it comes to the point where the relationship is flipped and it's the Doctor impressing her with how much he knows about her intimin secrets and repeating her "Spoilers!" catchphrase to her to keep her from learning things about him too soon. If the Universe doesn't actively prevent changes to the time line, expect to see some form of Time Police who's job is to track potential changes to the time line and prevent them from occurring. This is best seen in the organization known as the "Time Variance Authority" (TVA) in Marvel's Loki series. Here the TVA has knowledge of all events in all of time and will respond to any potential changes by removing branching from the time line from forming. Here, it's actually possible to change events through time travel (or rather, create alternative parallel universes) but an intelligence is preventing the change. Many Butterfly Effect stories are actually set in such a way where the Protagonist is charged with time travel missions to prevent the bad future from coming to pass by stopping an antagonist time traveler causing the bad future. In other works, the time travel changes were based on an incomplete understanding of the past events. In Futurama, Fly is under the impression that The Grandfather Paradox is in play and then accidentally kills his Grandfather before he has sex with Fry's Grandmother. Believing that the Universe is going to unravel because of his mistake, Fry throws all the rules of caution to the wind and sleeps with his Grandmother... only here, Fry realizes the man he thought was his Grandfather was not his biological Grandfather... Fry's own... "Past Nastification" resulted in the conception of his father. The drawbacks to this type of paradox are less physical and more philosophical: First, if something from the future changes the past in such a way that it causes the events of the future to result in the time travel, there is a fundemental break of the law of conservation of matter, since in effect the stable loop of time creates itself (and leads to the other name for this to be "The Bootstrap Paradox" as, like a self-made man, the cause and effect of time travel being the same event means that the product of time travel is self-made. However, the philosophical ramification of this means makes a scary implication that we have no free will. The reason for Loki's antagonism towards the TVA is, as a trickster, he is all about Chaos, which cannot be controlled, but the TVA's correctios means that only the Chaos they permit can exist... which isn't chaos at all, but order made to look like disorder... if we make the same choices every single time we play through the time line, than do we have free will, or just the illusion of free will.
