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
Nobody Loves Me, Everybody Hates Me, Guess I'll Go Eat Worms
When Howl's Moving Castle begins, Sophie Hatter believes all of the fairytale clichés she has been told about the eldest child of three being doomed to failure. Even though she loves her sisters, she's getting a good education, and she gets along well with her stepmother Fanny, it just seems inevitable that her youngest sister Martha is going to go off to find fame and fortune and Sophie will be lucky with an ordinary, steady-paying job.
So Sophie settles for a position as an apprentice in her family hat shop, even though she finds it boring, isolating, and depressing. She starts talking to the hats that she makes, which seems to have an odd, even magical effect on them (although Sophie doesn't appear to notice that her talking has any kind of power). The only thing that shakes Sophie out of her depression (oddly) is the arrival of the Witch of the Waste, who transforms her into an old woman to get rid of the competition.
Rising Action
Being Old Is the Best Thing That's Ever Happened to Me
Sophie finds old age surprisingly liberating—she's a lot less nervous about life in general now that her fears about failing because she is the eldest daughter appear to have come true. She doesn't need to worry about it anymore. Still, Sophie doesn't want her family to see her like this, so she leaves Market Chipping.
Sophie spots the moving castle of the Wizard Howl running close by and decides to hop aboard. Sure Howl has a reputation for eating girls' hearts and stealing their souls, but Sophie isn't a girl anymore, is she? Inside the moving castle, Sophie meets Howl's apprentice Michael and the fire demon Calcifer. And that night, as Sophie sits by the fire, she and Calcifer make a bargain. Calcifer can see that Sophie is under a curse, and he offers to break it for her—but in exchange Sophie has to end the contract binding Calcifer and Howl. Sophie agrees, and decides to stay in the moving castle.
Climax
Howl the Slippery Slitherer-Outer
The next day Howl himself appears. Sophie announces that she is going to be his new cleaning lady, and Howl doesn't say no. He doesn't say yes, either. In fact, Howl hates to commit to anything. And this trait of his personality keeps Sophie bickering with him for the next six-odd chapters, from Chapter Four to Chapter Ten. Sophie sees that Howl is a fickle, unreliable, overly dramatic baby. But she also sees that he can be unexpectedly kind—he never brags or draws attention to the decent things he does, like selling his spells for less than their value to the poor people of Porthaven.
But this slow process of settling in to life at the castle—cleaning, selling spells, helping Michael with his attachment to Sophie's youngest sister Martha, doing her best to prevent the bond between Howl and her middle sister Lettie—suddenly takes a turn when Michael starts practicing a weird spell he thinks Howl left for him. The spell is actually a lost English assignment that came from Howl's nephew Neil.
It turns out that Howl is from our world, and that the Witch of the Waste has somehow used her discovery of his origins to lay a curse on him from Wales. Neil's English teacher Miss Angorian is both (a) lovely, and (b) very interested in this whole spell business Howl's got going on. Sophie despairs that Howl is going to break Lettie's heart and hook up with Miss Angorian to fill the time before the Witch's curse finally takes action. Now that the curse has found Howl, it's just a matter of letting the impossible things in the curse come true, and then it will be too late for Howl.
Falling Action
When Is That Curse Coming, Again?
One of the handy things about the fact that the Witch's curse is not immediate and needs a little time before it takes hold on Howl is it gives the book a chance to ramp up suspense. We know we're approaching some kind of showdown with the Witch that is generally terrorizing the area (and who specifically transformed Sophie into an old woman), and we're just waiting to see how it happens.
Of course, even though we're basically just waiting for the clock to count down, there is still stuff to do: Howl has (extremely reluctantly) become the King's Royal Magician, so it's supposed to be his job to find Prince Justin, the King's brother who has disappeared. Sophie sees no signs that he is actually doing this, but he does disappear everyday—Sophie assumes that Howl is courting Miss Angorian.
Sophie also adopts a dog who turns out to be a man under enchantment. This dog-man is in love with Lettie; she's the one who sent him to look out for Sophie at Howl's castle. Apparently all of Sophie's friends and relations actually know that she is under a curse—as does Howl. All of this time, Howl has been running around trying different methods of taking the curse off Sophie without her knowledge.
Sophie is furious that Howl has been organizing so much behind her back, but she also seems to be angry that, in her current elderly state, she can't be with Howl in the way that she wants to. (At last—romance.)
Resolution
Lesson Learned: Avoid Falling Stars or You Might Lose Your Soul
When Sophie hears that the Witch of the Waste has (supposedly) taken Miss Angorian hostage in her fortress to draw Howl out one last time, she immediately uses a pair of magic boots to run over there. After the Witch takes Sophie captive, there is a showdown between the Witch, Howl, and a scarecrow (who is also an object of the Witch's curses). The Witch dies surprisingly quickly, which is a sign, Howl says, that her fire demon had almost taken control of her. And who is her fire demon? Miss Angorian.
So here's the final situation: fire demons are falling stars, right? Howl caught one of these stars several years ago. This star was frightened because when a star falls it knows it's going to die. So Howl made a deal with this star: he gave the star—Calcifer—his heart so that Calcifer could keep living, but Calcifer has to keep loaning Howl his magic in return. The Witch and Miss Angorian made a similar kind of deal many years ago, and it totally corrupted both of them. In other words, giving up your free will can make you feel trapped at best and like less of a person at worst.
Back in the present Sophie uses her newfound talent for talking life into things magically to give more life to Calcifer once she pulls Howl's heart away from him. Sophie restores Howl's heart to his body, freeing him from his contract with Calcifer, and Howl takes the Witch's withered, dried-up old heart from her fire demon, thus killing Miss Angorian. All of the Witch's curses (including the one keeping Sophie old) have now been broken.
Howl decides that it's time for him and Sophie to "live happily ever after" (21.111). So even though Sophie has been feeling this giant sense of doom because she's the eldest daughter of three, she actually gets her fairytale ending.