Continuity is a word used when talking about art, especially serial art. Serial art is when an artist paints several pictures which all have something to do with each other.
Pop culture [change]
The six Star Wars films tell one long story which continues from part to part. This is where the word continuity comes from. Continuity refers to consistency from part to part-- that is, everything should fit together like one whole piece.
At the end of the Star Wars film The Empire Strikes Back, Luke Skywalker loses one of his hands. In the next film, Return of the Jedi, he has a fake hand to replace the one he lost. If he had his real hand again, it wouldn't make sense. The audience would say, did not he lose his hand in the last one? Continuity is making sure things make sense.
Sometimes the word is used to refer to any work of art, like just one film. For example, a woman has a full glass of wine in one shot. In the next shot, the wine is gone. In a third shot, she has a full glass of wine again. That scene would have very poor continuity: it does not make sense, and it reminds the audience that they are watching a movie. Poor continuity is continuity that does not make sense and usually has mistakes.
In American comic books, many different characters live in the same world. For example, Superman and Aquaman live in the same world. But if Superman goes to Atlantis and it's different than Aquaman's Atlantis, it does not make sense. If they are in the same world, Atlantis should be the same no matter who visits it.
Making sure that all the different stories and characters fit together is working with continuity. Sometimes, the word is used to talk about the stories themselves.