Science fiction authors have been speculating about this for hundreds of years. Only recently has it become popular to compare and contrast different ideas to try and formulate a set of ideas that could plausibly work together with internal consistency and at the same time allow for interesting time travel stories of the sort we are familiar with.
There are lots of problems that can pop up in time travel stories. Logical inconsistencies in how the story treats the occurrence of time travel is a major one. Anyone who has thought carefully about the movie Back to the Future has noticed that in the first movie Marty must be careful not to alter time too much or else he might cause himself to cease to exist, or worse, disrupt the fabric of the entire time space continuum in an apocalyptic paradox. However, in the sequels he discovers there are multiple timelines and apparently the universe is a bit more resilient than Doc Brown originally thought, leaving you to wonder why he was in danger of fading from existence at all in the first movie? And just how much time does a time traveller get to fix time once he has mucked it up a bit?
It would seem that if you could travel through time and change events that already happened, a lot of paradoxical situations can result that can't be easily explained.
The first, and most obvious paradox, in my opinion, is that if you went back in time for a particular purpose, and you achieved that purpose (let's say you killed Hitler’s father to prevent his birth and presumably World War II), history would change and you (the new you in the changed universe) probably wouldn't have felt it necessary to go back in time for that particular purpose anymore (it having been achieved by the old you already) and so there would be a different you who would presumably go back in time for some other purpose and so on repeating infinitely with infinite time “clones” going back in time for various different reasons (the previous reason no longer necessary in the next iteration).
Where does the original you fit into this picture? Do you cease to exist to be replaced by the new you? Do you continue to exist alongside the new you? Could you have even changed time in the first place if by doing so you caused the events and purpose of your own time travel itself to change?
Also, consider that if you clearly are not present in history, and you traveled in such a way that you inserted yourself into that history, that clearly is not the same history, but a new one with an entirely (or at least slightly) different chain of events.
Now, complicate matters by having other time travelers jumping around chasing each other and killing each other's grandparents (or their own)—hence the infamous "Grandfather Paradox." How can you tell what history would look like if multiple time travelers are interacting with each other in ways we can only begin to fathom? There seems to be as many ways time travel could work as there are storytellers coming up with time travel stories.
Because there are a lot of ways in which time travel could work, many of them are inconsistent with each other. This can be particularly embarrassing if you create a story that is inconsistent with itself.
But can we imagine a logical set of rules for time travel that would be fairly predictable and consistent for all participants in the story? If we want a time travel story to sound plausible we should spend at least a little time thinking about it.
Question: What happens when you go back in time?
Answer: History is presumably just as it was before you went back in time, with one notable exception. You (including your time travel device, should you need one, plus your possessions, and your time traveling companions, and perhaps some nearby matter and energy) are inserted into that history.
Question: What happens in the future I left behind if I change something?
Answer (a): What if there is only one static unchangeable timeline?
Sorry, you can’t change anything. Perhaps it's impossible to physically exist in the past, you can only look around. This is nothing more than remote viewing, no actual time travel. This seems like a very boring story for time travel but perhaps the most realistic.
Or stranger yet, try as you might, you can't change history. It turns out you were here all along. If you paid close attention to your history books and old newspapers you may have already seen yourself or the effects of your time travel because it's always been there. It would appear that you have no free will as you are destined to do exactly as you always have done no matter how much meddling you try to do. No free will makes for a very anticlimactic story and a very unsatisfying theory of how things work in physical reality. However since we can't necessarily prove free will exists or doesn't without some kind of time travel this is still a viable option for some.
Another alternative is to say that you can change minor things, but in the end everything works out to being pretty much the same anyway. This seems equally unsatisfying to me both as a story mechanism and as a description of the physical mechanics of time travel.
Answer (b): What if there is only one timeline but it can be changed somehow?
If you change the past the effects ripple forward in time resulting in perhaps small or perhaps devastating consequences depending on how sensitive the physical world is to change. Step on a butterfly and suddenly all of the bets are off at the horse races, and every other seemingly random event starts going haywire causing the world to become an entirely different place almost instantly. Or perhaps the world is surprisingly resilient to change. Killing Hitler only results in some other dictator rising to power to have virtually the same effect on history
The consequences might affect the time traveller himself, causing him to cease to exist, or to change into a new person with different experiences and memory. Perhaps he exists in between somehow with overlapping memories of history or shifting physical traits. Perhaps he slowly begins to fade away if he has nullified his own existence.
The consequences of changing time might take some time to happen (how much time? Only the author of the story can decide that!). Perhaps the consequences occur instantaneously or perhaps the traveler is safe until he catches up with the moment of his birth or perhaps the time of his original time travel and then he must pay the piper. Perhaps the consequences race forward at days or months per second and he must race against time to fix any blunders that may adversely affect him or his loved ones. The possibility of getting the traveler or even the whole universe in a never-ending loop, or perhaps causing all of reality to break in an apocalyptic or paradoxical blunder must be avoided. Most time travel stories fall in this category. There are so many ways a single time might respond to changes that no two authors ever seen to imagine things in the same way. These can make satisfying stories but if you nitpick enough you're almost certain to find inconsistency and contradiction about how time travel physically works between stories and often within the same story.
Answer (c): If an additional but temporary timeline is created.
Nothing happens in the future. Your actions only ripple forward in a temporary timeline copy while you are enjoying your time trip. Change whatever you like to your heart's desire. When you return home (assuming that's possible), all of your changes (with the exception of your aging) pop out of existence as though they never happened. If you can bring objects or people to the future, they are exact replicas or clones, their historical existence is entirely different. Future excursions to the same past time are possible and you can redo as many times as you like without having to worry about your previous trips because poof, all those changes cease to exist as soon as they become unimportant to the story. This kind of time travel is also unsatisfactory as it limits what time travel is capable of and there doesn't seem to be any real physical reason why it should work that way other than because it prevents the storyteller from accidentally creating complication. Still superior to single unchangeable timeline with “no free will” in my opinion.
Answer (d): If an additional persistent timeline of causality is created.
Nothing happens in the future you left behind. Your alterations ripple forward in a different causality totally independent from the one you left behind. It is entirely possible that you can never return, as travelling forward in time will move along the new chain of events which may be altered significantly or slightly from when you left. If you are able to return back to your original timeline of events somehow (presumably only allowed after the moment you left, lest you create yet another timeline in between) you will discover both timelines exists independently from the other. It would seem that any sort of time travel would create new causality timelines so long as previous events are to changes (the act of time travel is a change in itself however insignificant). Presumably there might be some way to identify various timelines and travel between them, but this opens a whole new can of worms about how time travel works.