Before we get into the "-quels," we need a base story. Let's call it "Story X." This will be what each "-quel" relates to throughout the post.
1. Prequel
The prefix "pre" means before, so this is a story that exists in the same world as "Story X," but it is set before "Story X." An example of this in mainstream media would be Star Wars Episodes I-III, which were created after Episodes IV, V, and VI were released.
2. Sequel
This is the term that people are most familiar with because a sequel is a story that occurs after an established story. "Story Y" takes place after "Story X," but it is connected to "Story X" through the characters, events, and/or settings. Star Wars Episode 5: The Empire Strikes Back is a sequel to Star Wars Episode 4: A New Hope. When a book or movie is very
successful, a sequel will often follow to allow readers/viewers more time with characters they have grown to love.
3. Interquel
"Inter" means "between," so an interquel occurs between two already-established stories. If "Story Z" follows "Story X," but there is room between the two for another story, you could add "Story Y" as an interquel. An example of this in mainstream media would be Black Widow because this story takes place after the events of Captain America: Civil War and before the events of Avengers: Infinity War even though Avengers: Infinity War had already been released by this time.
4. Inquel
An inquel is a story that takes place within a gap of time in another story. If Story X has a period of time that is rushed through without being fully explored, that would give an author the opportunity to later go back and write an inquel. These are not common, but an example of this would be Bambi II, which occurs during the established events of Bambi. It explores the time period when Bambi is still a fawn and his mother has just died. It ends prior to the time when Bambi is shown as an adult in the first movie.
5. Paraquel
A paraquel isn't something you come across very often, but this is a story that occurs within the same world at the same time as another story. This means you will likely see overlapping characters and scenes, though they probably won't all be the same. If an author writes a story of the same events from another character's perspective, that would be a paraquel. If "Story X" and "Story Y" are happening at the same time to two different characters within the same world, the two stories would then be paraquels of each other. My book, The Heart of Everton Inn, is a paraquel in relation to The Secret of Drulea Cottage because the two stories are taking place at the same time, but The Heart of Everton Inn features Adaira Stubbins as the main character, whereas Briony Fairborn is the protagonist of The Secret of Drulea Cottage.
6. Circumquel
This type of story has events occurring before "Story X" and events occurring after "Story X." I can't recall ever watching a movie or reading a book with this type of story, and it seems like it would be very difficult to create a compelling one. I would think that making two separate stories (a prequel and a sequel) would be a better stylistic choice than putting both past and future events into one story.
Suffice it to say, there are many options when it comes to how stories relate to each other. I hope this helps you clarify the names a bit!