Fictional universes, continuities, mythologies, and canon
In SF and fandom, any work might be retold numerous times, leading to numerous different takes on characters, settings, and themes.
Some common retellings include (a) a creator who revisits a world and tells a story again from a different perspective, or expands it in some way, providing variant or alternative details; (b) successor creators / rightsholders, particularly within media SF (Star Wars, Star Trek) or popularly successful literary works (Gone With the Wind, Flowers in the Attic) may "authorize" sequels or new entries in the "franchise"; (c) creators regularly re-work existing tropes, myths, stories, and legends, into new works of their own vision -- short informally published works are usually described as "fan fiction" (which has its own genre conventions), while more formally published works may be described as parody, pastiche, homage, etc.
SF works that involve detailed world-building often encourage passionate devotion and a significant participatory culture, which can manifest in fan fiction and fan sites dedicated to producing reference material about the fictional universe.
In looking at the characters and world-building -- the elements of the fictional world, such as the setting, including historical backstory, place, time, and events -- numerous retellings present a problem for anyone attempting to rationalize the world into a cohesive, consistent story. We briefly discuss and define below some concepts useful for understanding the problems; each term may overlap with other terms, and is fuzzy on its borders.
Continuities
Fictional continuities are basically individual fictional universes in which characters and stories exist. They can include multiple timelines, parallel universes, and so on, but within the constraints of that world there is some element of continuity -- often the characters; less often species, planets, or others.
Some fictional continuities might be entirely in print (a in Category:Valdemar, but others might span multiple media. For instance, a single movie with action figures; or, a whole series of related works in different media, like novelizations, tv series, books, video games.
Fictional continuities can have internal continuities, alternate continuities, and discontinuities. For instance, a tv series may give different versions of the backstory. Or there may be different versions of a story, as in Buffy the Vampire Slayer, the movie, and the TV series, which are somewhat different; or Battlestar Galactica, the 1978 and 2004 versions. When people distinguish between "canon" and "non-canon" among authorized works, they are determining that some aspects of the fictional continuities are so discontinuitous that they cannot be rationalized into one consistent narrative history. For instance, some people describe the Buffy movie as "non-canon" for the Buffyverse, because it is so inconsistent with the Buffy & Angel TV series and other later works.
Comic books are known for creating new "continuities" for their characters. In one continuity, for instance, there may be character deaths without resurrections, while in another continuity a particular character may never die or may be resurrected. Still, all the comic book continuities can be grouped into super-continuities: All X-Men comics, for instance, have some elements of continuity (usually some characters and some aspects of the backstory/mythology) that are not shared with, for instance, the Buffyverse.
See retconning.
Canon
Canon in this context describes and differentiates between officially authorized works that form a coherent and consistent structure. Officially authorized works may be later deemed "non-canon" if they conflict with the rest of the story -- for instance, Alan Dean Fosters' Splinter of the Mind's Eye, an authorized original novel in the Star Wars series that was deemed non-canon after release of later films in the series made a suggested relationship between Luke and Leia canonically undesirable).
Mythology
The mythology of a world is the invented backstory, particularly the SFnal elements.
A world's mythology can look like real-life mythology. In J.R.R. Tolkien's Lord of the Rings works, for instance, the mythology includes a fictional mythology, involving stories, fables, and tales of earlier magical beings and conflicts. The creation of fictional mythologies that operate as a mythology within the fictional world (ancient stories, oft-told, half-remembered, with ambiguous historicity) is called mythopeia. It may be that mythologies are a subset of mythopeia.
However, "mythology" can also refer more generally to the SFnal elements of the world's backstory, or the unfolding of the story arc of a work, as in the myth-arcs (mytharc) of The X-Files, Babylon 5, and Buffy.
Franchises
A franchise is the collection of works for a particular universe. Used generally, it simply means the collection of works. However, it typically is used to refer to the authorized (by the copyright or trademark holder) use of the characters and settings of a particular work or series of works. Franchises typically revolve around media works, from film, TV, or video games, and may include novelizations, translations and adaptations, animated versions of the works, soundtracks, action figures, other collectible items, various distributions and formats of the works, and new texts or films in the franchise or series.
Fictional universes
A fictional universe is similar to the narrower version of the "continuity" concept. It may be used to refer either to a physical universe, as in a place -- for instance, the "Star Wars" universe would include the various planets and history within the galaxy "far, far away".
The "fictional universe" may also refer to the universe/setting defined by continuing characters within a work. For instance, in Robert A. Heinlein's The Number of the Beast, characters go jaunting through multiple parallel universes; nevertheless, it is clear that they inhabit the fictional universe created by that work, and not the fictional universe inhabited by, say, Tenar in Earthsea.
Characters may breach their own fictional universes; this is referred to as breaching the fourth wall, speaking to the audience or acknowledging the fictional nature of their own universe.