Continuity used in its most basic form is simply consistency. In creating any form of media, whether it be animated, written or live action, it is rarely done in perfect chronological sequence. Scenes are written, shot or animated in a different order than it appears to us. There is an entire production manager whose job is to maintain continuity from one place to another.
This often involves little things, such as how much water is in a glass from beginning to end of a dinner scene. Or the director decides to try a new take with a character standing in a different part of the room, their job is to make sure the audience doesn't catch the gaffes and slip-ups associated with this process.
On a broader scale, continuity is also about consistency with the dialogue and story. Cuts and edits may take out entire sub-plots of a film, but if a joke remains that depended on something that was edited, then the joke will be lost to the audience.
Continuity can refer to backstory. Did the character learn their skills from a Pirate or a Ninja? The dialogue tells us two different stories. To help with such things a Universe Bible is often created and held onto, just to avoid those mistakes that the fans will catch. This is the same thing regarding Canon, but in a much larger scale.
As a countable noun ("a continuity"), "continuity" is often used to refer to a set of stories that are mutually Canon, a sort of self-contained possible world. (Compare and contrast with The Verse, where not everything has to be canon.) For example, a series would typically have its own continuity, and sometimes two series by the same author are in the same continuity, so that what has happened in one is canon in the other; but if they aren't, it's not. In other words, a continuity is that inside which there needs to be continuity, and outside of which there doesn't. It follows from this that in some cases it's not possible to tell whether the word is being used like a count noun or not, but then it makes no difference for meaning anyway. Sometimes a new adaptation contradicts old canon and thus happens in an Alternate Continuity of its own.
The count noun use possibly started out from a use of the word "continuity" to refer to a Story Arc between issues of superhero comic books in the 1960s. As the stories got more complex, the term's meaning evolved to mean the combined histories of all the superheroes by a given publisher.