Character Analysis
Martha is Sophie's youngest sister, and she's at least as opinionated as Lettie. However, unlike Lettie, her ambition is not to become rich and powerful—instead she wants to marry, settle down, and have ten children.
With this goal in mind, while Martha is working as Mrs. Fairfax's apprentice she learns enough magic to change her appearance to look like Lettie's. And then she and Lettie switch places: Martha becomes an apprentice at Cesari's pastry shop and Lettie becomes Mrs. Fairfax's student. Martha only comes clean to Sophie about this switcheroo once Sophie finally gets up the courage to leave her hat shop and visit the pastry shop (before she turns old, of course).
Both Lettie and Martha refuse to give in to their proper fairytale roles: even though, as the youngest daughter of three, Martha is the one who should be going off to have adventures, Martha knows that's not what she wants. Sophie is the only Hatter who clings strongly to the fairytale clichés, and it's part of her journey to become more like her sisters, since they agree that she is "far too clever and nice to be stuck in that shop for the rest of [her] life" (2.16). Sometimes younger siblings really know what's up.
Rage
Martha has two traits the turn out to be important in the novel. First, she is clearly angry at Fanny—her mother but Sophie and Lettie's stepmother—for sending Lettie to the pastry shop and Martha to Mrs. Fairfax without listening to their wishes. She's also furious that Fanny has been keeping Sophie as an apprentice in the hat shop when it has so clearly been bad for her. Martha communicates her frustration to Sophie:
Mother knows you don't have to be unkind to someone in order to exploit them. She knows how dutiful you are. She knows you have this thing about being a failure because you're only the eldest. She's managed you perfectly and got you slaving away for her. (2.20)
However, when Sophie meets up with Fanny later on in the novel and Fanny is so happy to see her and has been so worried, Sophie realizes that Martha's view of her mother's motivations are just a tiny bit biased. Martha is angry at her mom, leading her to interpret her behavior in the worst and cruelest ways possible.
Indeed, looking back on this time in the hat shop from the perspective of Old Sophie, she can see that Fanny was also struggling with unhappiness and frustration following Mr. Hatter's death. Maybe she could have paid more attention to Sophie, but she isn't deliberately forcing Sophie to "slave away for her." Sophie realizes that she listened to Martha's point of view as though it was complete truth without considering all sides of the issue, which wasn't fair either to Sophie or to Fanny. Slowly Sophie is learning to be a better, fairer judge of character.
Romance
Beyond Martha's resentment toward her mom, the other character trait that makes her important in the novel is her desire for love. Martha places herself at the pastry shop after magically changing her appearance so that she can find a man who will love her for her rather than for her disguise as the spectacularly beautiful Lettie.
And indeed Martha does find the perfect man for her: Michael Fisher, Howl's apprentice. Their romance seems sweet but not exactly the stuff of legends—they're both fifteen, which is too young to get married, so they are waiting around until they are ready to start their lives together. They are both steady and responsible people who seem fairly certain about what they want out of life, and they both seem pretty devoted to one another.
Martha and Michael's relatively stable relationship stands in strong contrast to the dramatic disaster developing between Sophie and Howl. We certainly never see any sign that Martha and Michael get into giant, over-the-top fights that produce green slime in huge gobs throughout the moving castle.
Sophie and Howl may be older than Martha and Michael, but they have the emotional maturity of kids sometimes. Still, would you read a whole book about the sensible courtship of Michael and Martha? Sophie and Howl's ridiculousness is what makes their relationship worth narrating; by contrast, most of what takes place between Michael and Martha happens offstage because it's definitely nice but also dull.