Calpurnia's hard hand comes into use when Walter Cunningham is invited to lunch on Scout's first day at school and Scout makes fun of his back-country ways:

Walter poured syrup on his vegetables and meat with a generous hand. He would probably have poured it into his milk glass had I not asked what the sam hill he was doing.

The silver saucer clattered when he replaced the pitcher, and he quickly put his hands in his lap. Then he ducked his head. Atticus shook his head at me again.

“But he’s gone and drowned his dinner in syrup,” I protested. “He’s poured it all over—” It was then that Calpurnia requested my presence in the kitchen.

Scout knows enough about the Cunninghams to know they are poor, but she doesn't really understand what that means. Her explanation to her teacher about why Walter did not have bring a lunch to school—"He's a Cunningham, ma'am"—shows that for her, the Cunninghams are a breed apart, obviously different from everyone else. Both Atticus and Calpurnia are upset over Scout's calling attention to Walter's difference, but of course Calpurnia's response is more pointed. As the black cook to white family, Calpurnia is in a difficult spot. She is aware of her inequality, yet she also recognizes the respect Atticus has for her, and she feels a genuine affection for Jem and Scout. Scout's behavior touches a nerve:

“Hush your mouth! Don’t matter who they are, anybody sets foot in this house’s yo‘ comp’ny, and don’t you let me catch you remarkin’ on their ways like you was so high and mighty! Yo‘ folks might be better’n the Cunninghams but it don’t count for nothin’ the way you’re disgracin‘ ’em—if you can’t act fit to eat at the table you can just set here and eat in the kitchen!” Calpurnia sent me through the swinging door to the diningroom with a stinging smack.

Scout complains to her father about this treatment, but Atticus will hear nothing of it: "I’ve no intention of getting rid of her, now or ever. We couldn’t operate a single day without Cal, have you ever thought of that? You think about how much Cal does for you, and you mind her, you hear?"

Later, Calpurnia "tries to make it up" with Scout, telling her she missed her while she was at school and making her crackling bread as a treat:

Calpurnia bent down and kissed me. I ran along, wondering what had come over her. She had wanted to make up with me, that was it. She had always been too hard on me, she had at last seen the error of her fractious ways, she was sorry and too stubborn to say so. I was weary from the day’s crimes.

Of course, Scout's analysis of Calpurnia's motives is exactly wrong. Cal may or may not have felt she had been "too hard," but what she definitely feels is love. She may have a hard hand, but she has a soft heart.

In this description, Calpurnia's character and history are revealed. The fact that her hand is wide and hard shows that she is no stranger to manual labor. When she spanks Scout, we see that she is also not afraid to use discipline when she sees fit.

In Calpurnia, Scout has a strong woman to balance her father's relative softness. Atticus is a man of the mind and of gentle reasoning. Calpurnia is hard as nails and a more hands-on (no pun intended!) disciplinarian.