When done well, time travel is one of my favorite devices in fiction. Watching a good time travel story coming together is always satisfying, like solving a puzzle. On the other hand, there are plenty of examples of awfully handled time travel. I've written up an explanation of all the types of time travel one might encounter in fiction along with some examples.
There are multiple types of time travel in different pieces of fiction. How one travels through time is not important, but the rather the rules that constrain actions when traveling and time travel's affect on the timeline are. Some people break these down into 3 categories others into 8. I like the 3 broad types as most examples of fall into one of the categories.
Fixed Timeline - No paradoxes occur. Something will always prevent you from changing a future event as observed events in the future cannot be changed. Going back in time simply sets events into motion that already happened in the future.
Dynamic Timeline - Anything goes when you travel back in time and paradoxes resolve themselves...somehow. In some cases, paradoxes actually don't resolve and in others you break the universe. Sometimes the timeline is 'sticky' and things just tend to go back to normal unless you take some drastic action.
Alternate Timelines - Traveling back in time actually sets you on a new timeline, therefore any changes you make do not cause a paradox. This can sometimes be called a Multiverse if one can travel between the alternate timelines.
One point I would like to add, which can apply to any of these three categories, is that some types only allow information (usually in the form of a person's consciousness) to be sent through time. If this is the case, I append the phrase "information only" to the type of time travel.
Diagram of the timelines in each type of time travel. The blue arrow represents an actor traveling through time.
A paradox is when an event occurs differently in the future due to changes made in the past. The most common one is the grandfather paradox, where one travels back in time, kills their grandfather before they were born, yet then could not have traveled back in time in the first place due to not existing.
Paradoxes only factor into the first two cases for time travel. Due to the nature of only having one timeline, paradoxes have to be resolved or prevented. In dynamic timelines, you can have all sorts of creative ways of resolving them from just letting them kinda happen to breaking the universe. The most common way to resolve these paradoxes is to allow the change to affect the future, but prevent it (or partially prevent it) from affecting whomever traveled back in time.
In fixed timelines, paradoxes just don't happen. This is really easy to pull off in movies or books as the author can just writes the story to prevent it. Consider the grandfather paradox; "The bullet misses his grandfather", "He shoots the wrong person", "The grandfather survives and meets his grandmother while recovering in the hospital." However, how would this work in a game?
Time Dilation - This is any case where time moves slower (or not at all) or faster for someone. While interesting, it's not time travel.
Foresight - Seeing the future through premonitions isn't really time travel either, but can often have some of the same rules. For example, often one seeks to avert a vision of the future but ends up causing it leading to a fixed timeline. In other's like a Christmas Carol, a vision of the future leads to that vision being avoided, like a dynamic timeline. However, if the information from the future is actually being sent from the future, then it becomes a form of information only time travel.
Any time travel that involves jumping between different times, but with no restrictions on what one can or can't do in not by itself an interesting concept. This case can be either fixed or dynamic timeline, but it doesn't really matter. It is usually setup such that the times you can travel to are so far apart as to make it impossible for actions in the past to influence the future. While this is technically time travel, it's not the time travel and rules around it that are interesting, but rather just the location and time. Since there is no possibility of a paradox. I'll class this as "fixed timeline - paradox irrelevant" as a fixed timeline is probably the default.
Harry Potter 3 - Fixed Timeline. A neatly written fixed timeline. Everything Harry observes the first time happens as his future self was there to save Sirius, Buckbeak, and even himself.
Tenet - Fixed Timeline. This one has a cool twist on how one travels through time. It's also my favorite time travel movie. While the people of the future believe they can change the future with some super weapon, there is no evidence of this and the timeline never changes.
Back to the Future - Dynamic Timeline. Perfect example of a dynamic timeline. Marty goes back in time but stops his parents from getting together and slowly starts to fade from existence. How then did he go back in time if he never existed? Great movie anyway.
Dr. Who - "Fixed" Timeline. I have watched enough Dr. Who to know that time travel is loosely explained ("A big ball of wibbly wobbly, timey wimey stuff"), but I'm pretty sure it's one timeline (ie no multiverses). However, it mostly sticks to a fixed timeline as they don't really change the future and are somewhat limited in what they can do in the past. However, some episodes lean pretty close to fixed timeline - paradox irrelevant.
Travelers - Dynamic(?) Timeline, information only. In this one, agents in the future send their consciousness back to people in the past. This one is tough to classify as our travelers take many precautions to avoid messing with the future...in order to avoid ecological disaster in the future. I'll call this one Dynamic Timeline, information only with rules.
Stein's Gate - Fixed Timeline, information only. While the show ends with our heroes avoiding a dystopian future, the main time travel we see in the show involves a fixed timeline.
Marvel Cinematic Universe (Avengers: Endgame, Loki) - Multiverse/Alternate Timeline. Time travel is simply solved by having a new timeline created when one goes into the past and changes things. It is even called a multiverse.
Groundhog Day - Alternate Timeline, information only. This one involves Phil's consciousness being unwillingly sent back to the beginning of the day. One could interpret the days that he lived over and over as alternate timelines.
The Time Machine - Fixed Timeline, paradox irrelevant. While not the first instance of time travel in fiction, this is probably the one that made the idea popular. Still, there isn't really any effect of past actions on the future, but the narrator does worry about traveling forward into a solid object.
While relatively easy to pull of in most media with a set story; books, movies, video game cutscenes, time travel as a mechanic in a video game presents us with several challenges. How does one prevent the player from creating paradoxes if paradoxes are not allowed? How does one resolve paradoxes if they are allowed?
However, there are several examples of time travel integrated into the gameplay mechanics of video games.
Braid - Alternate Timeline, information only. If there was one "time travel" video game it would probably be Braid. The time travel in Braid doesn't fit neatly into any of the 3 categories. When you travel back in time you actually rewind the entire world, so one could see this as sending ones consciousness back. There are several gameplay mechanic twists one of them being some objects that are immune to time being rewound. To make things more complicated, there are other mechanics in the game that involve time but aren't time travel. I'll put this one loosely in Alternate Timeline where only information can be sent back as we can think about rewinding the world as sending your conscious back in time to a new branching timeline.
Titanfall 2 - Fixed Timeline paradox irrelevant. This level in one of my favorite games has a mechanic that allows you jump between future and past. While this level is probably one of my favorite levels ever, its more of a platformer as the past and present are so far apart that you have minimal influence on the present in the past.
5D Chess With Multiverse Time Travel - Multiverse/Alternate Timeline. It says it in the title, this one is squarely (heh) in the alternate timeline category. It even has well defined rules for moving between timelines. While the game's title sounds like a joke, the rules are logically sound. Whether or not a regular human can follow them is another question.
No two examples of time travel are identical even if they fall into the same category. In fact, time travel rules often don't remain the same within a universe. Harry Potter and Cursed Child features Dynamic Timeline time travel and the Terminator series seem to change the rules of time travel with every movie. However, by having some broad categories, we can get a general idea of what readers or gamers expect when presented with time travel.
You might have noticed the focus on video games. I've been fascinated with the idea of a Fixed Timeline time travel video game for some time. If you're interested, I've created Time Travels, a game that explores this concept.