Most good stories start with a fundamental list of ingredients: the initial situation, conflict, complication, climax, suspense, denouement, and conclusion. Great writers sometimes shake up the recipe and add some spice.
Exposition
Orphans Unite
Reynie (an orphan), Sticky (a runaway/orphan), Kate (an orphan), and Constance (a gorilla—just kidding, she's an orphan, too) all pass several difficult tests which qualify them to work with four tremendous adults (three of whom, at least, were once orphans) on a dangerous mission to save the world from an evil genius with a sinister plot. We get everyone's backstories here—at least, everyone that can remember their backstory—and by the time their done filling us in, we also know why they've come together and what they hope to accomplish. More or less.
Rising Action
The Mission: Impossible
The kids get some training and some transportation to Nomansan Island where they find out just how sinister Mr. Curtain's sinister plan is. They find clues, decode messages, get really good at cheating, have more than a few narrow escapes, and eventually get close enough to Mr. C and his Whisperer to feel like things are really hopeless. Still, even after Milligan is caught, they proceed with their "bold, ill-formed, and likely to fail" (33.1) plan, which eventually puts all four of them face to face with big bad Mr. C and his big bad modded hairdryer (a.k.a. the Whisperer).
Climax
Constance is Especially Contrary
The kids are in the Whispering Gallery, and they've knocked out Mr. Curtain twice, but they still can't figure out how to disable the Whisperer. Worse, Mr. Curtain has weaseled his way back under the red helmet and brains are about to be swept. The tension is high, so when Constance climbs into the Whisperer and those wrist cuffs snap shut, we know the climax has arrived. If Constance can hold out, they'll win. And if Constance can't hold out, who possibly can?
We're definitely teetering on the totter of this turning point, and we're super relieved (and rather amused), when we realize, along with the others, that Constance could have freed herself at any time.
Falling Action
A Family Here, A Family There, E-I-E-I-O
Mr. Curtain escapes—but the people we care about get off the island, the Whisperer is defeated, and everyone who was previously orphaned or afflicted with amnesia now has a family. This is all falling action because even though some of the news is kind of exciting (Miss Perumal is adopting Reynie and Sticky's parents really do love him), it doesn't increase the tension at all. In fact, it decreases tension because it lets us know everyone is going to be all right.
Resolution
They're Still Children
There's a moment on Nomansan Island right after the kids encounter Mr. Bloomburg/Harry Harrison and realize that Mr. Curtain has been erasing people's memories, when the members of the Mysterious Benedict Society get a little depressed. As Trent Lee Stewart writes, "Now that the children finally knew some things, they all rather missed not knowing them" (20.1). When we read that, we worry that they've lost their innocence to some degree, and perhaps they have.
But the ending scene when the kids all go out for a snowball fight, we know that they're going to be fine. Sure they've seen some stuff, but they're still young and innocent. As Mr. Benedict says, "Let them play. They are children after all" (39.87). And that makes us happy.