The way reality (and by proxy, stories) works is that if one thing happens then something else does. Because of this, we have a natural sequence of events that happens. It’s a consistent sequence of events that have bearing on each other.
Man, describing continuity is difficult.
Basically, if something happened, it happened. Events that happen influence the next one. Yet how much this affects the story depends on, well, the story.
Let’s take The Avengers, because I love that movie and it has an example. Iron Man, Iron Man 2, The Incredible Hulk, Captain America, and Thor all lead up to the movie. Events in The Avengers reference what happened in ‘prior’ movies and hew to characterization established thus far. Thor’s worried about Jane Foster and Tony Stark doesn’t really trust Natascha Romanov all that much. Loki’s also got some issues to work out with kingship, sonship, and all that.
Thing is, it’s not so interwoven that you absolutely have to see all five movies to be able to ‘get’ The Avengers. It stands alone just as well as it stands as a part. Each character is still introduced and established. Watching the other movies adds to the experience, but you don’t have to. The continuity’s there, but it’s not restrictive.
Examples of this loose sort of continuity (events don’t contradict each other, but you don’t need to be a guru on the work to know what’s going on) abound. During Firefly’s brief tenure it would introduce a character or place and bring it back later. Saffron was introduced, then we meet her later as Bridgett (and then Yolanda [or Yo-Saff-Bridge for short]). Malcolm Reynolds instantly recognizes her again, of course. That’s continuity!
Or the Uncharted games. Each successive one builds on what’s been established earlier, but, again, one doesn’t have to play all of them to get the plot. Nathan, Sully, and Elena are introduced each time as is their standing with each other. They have a history (some of which we know about) that influences their actions. Playing the other games adds, certainly, but it’s not necessary. The plot doesn’t contradict itself but it’s still accessible no matter where you start.
Then you’ve got the opposite end of the spectrum. Lost’s continuity is so deeply, heavily interwoven that missing an episode leaves you trying to figure out what you missed. This isn’t necessarily bad, just not the most viewer friendly way to do things. Just about every event in Lost has connection and meaning that will pay off later. A seemingly-trivial event that happened once actually has deep repercussions, something that wouldn’t work as well were it not so tight.
So if you don’t watch Lost since the first episode you’ll be lost. Crap. We get that, so what else? The story seldom, if ever, contradicts itself. Events impact the next. Even when time travel gets introduced it’s done in such a way that doesn’t create gaping loopholes. Though time doesn’t always flow linearly in Lost, it doesn’t go back on itself. Storyline contradictions break the suspension of disbelief, leaving the audience thinking “wait, what?” instead of focusing on the plot. Lost does no such thing.
Continuity, no matter the amount, is always important. In a sequel we want to see what happens next to the characters and events given to us in the original. Pulling a Revenge of the Fallen and deciding to undo a lot of what happened in the first leaves a very sour taste in the audience’s mouth. The Dark Knight brought Batman Begins to its logical next step without blocking out a new audience. Toy Story 3 acknowledged all prior events but told an independent story (that didn’t tread on the feet of the first two). Don’t undo what’s been done.
Going all out works in some cases, in others it’s best to keep it light.
Just don’t end up like Metal Gear Solid and reveal in the fourth game that half the exposition thus far has been lie after lie after lie. Sure, it works as a twist, but it’s kinda confusing. Tell a story, and tell it consistently.
Also: buy my book In Transit! It’s got continuity in it, at least as much as you can have in a short story collection!