Norrington's Establishing Character Moment. He's a gruff Reasonable Authority Figure who warns Elizabeth that pirates are not all that cracked up to be, as she may assume from her books. Her father is concerned about Norrington scaring or fascinating her, but Elizabeth reassures Norrington she's happy to learn. When she sees Will in the water, he takes charge to save the boy. Norrington is relieved that Will is still breathing and orders him taken below to get medical care.
Elizabeth as a child saving Will's life. She sees a boy floating in the water, as part of a shipwreck, and alerts the crew. They get him on board, as he deliriously gives his name and thanks to her. Governor Swann orders Elizabeth to take care of the boy. Elizabeth finds a pirate's coin on him and hides it from the crew since she doesn't want a boy her age to get hanged. It's implied that Will fell in love with Elizabeth because he remembers that she saved him. She also keeps the coin as an adult, even hanging it around her neck.
While sailing into Port Royal, Jack removes his hat in respect for the hanged pirates he passes.
Norrington saying out loud that asking Elizabeth's hand in marriage makes him nervous, is Heartwarming combined with Funny. Especially since everything else we see from him is being a firm, strict and upright military man—this proves he really does have a softer side.
Jack saving Elizabeth from drowning at the start of the movie. He has literally no selfish reason to at all, in fact it slows him down on getting what he wants, and he doesn't even know her or that she's the governor's daughter. He just sees that she's in trouble and jumps in to save her without a second thought. Though his silly antics worked as an Establishing Character Moment for the classic nonsensicalness that defines him, this acts as an Establishing Character Moment to show that deep down (and though he'd never admit it), he is a good person.
When Elizabeth first falls, Norrington immediately takes off his jacket to dive in after her. He has to be physically stopped by the rest of the men nearby because the height and rocks would likely just kill him.
After Jack saves Elizabeth, Murtogg and Mullroy immediately rush in to help pull her onto the pier. Even if they're a bit incompetent, they still want to help save this woman.
Elizabeth and Norrington's relationship in this movie; while she loves Will, she considers Norrington as a family friend. What's more, she knows that he is a good man and trusts him. Norrington is even apologetic about arresting Jack despite Elizabeth's protests, because he understands her logic.
In a small way, Jack revealing that he knew Will's father, giving Will the first truthful description of Bill Turner he's ever heard.
Jack: Good man. Good pirate. I swear, you look just like him.
When Pintel and Ragetti take the time to tell a jailed up Will more about his father. They could have ignored Will's question, but instead they take the time to tell Will about Bill. Comes to show how much Bill is respected among the pirates.
Jack's drunken monologue to Elizabeth.
Jack: Wherever we want to go, we go, that's what a ship is ya know. It's not just a keel and a hull and a deck and sails, that's what a ship needs. But what a ship is, what the Black Pearl really is... is freedom.
When the prisoners in the cell next to Jack escape after a cannonball blows the wall open, one of them lingers long enough to express genuine sympathy for how Jack can't escape with them.
When the Marines are about to board the Dauntless, Murtogg and Mulroy share a brief handshake before they jump into the fray.
After Barbossa is shot and Elizabeth and Will had reunited.
Watch Norrington during Jack's hanging—he's completely stone-faced, and when Elizabeth says it's wrong and her father responds, Norrington doesn't say a word, but just looks at his boots. He agrees with her, but as Swann says, "Commodore Norrington is bound by the law."
And during the same scene, the simple sight of Cotton's parrot, foreshadowing that they've come back for Jack after all.
Jack: I though you were to keep to the Code? Gibbs: We figured they was more like guidelines.
Near the end:
Commodore Norrington: You forget your place, Turner. Will Turner: It's right here, between you and Jack.
Will calling Jack a "good man" after his failed rescue attempt. Considering everything the man has done (and put Will and Elizabeth through) that's saying a lot.
Jack himself was so flattered that when it came time to say goodbye, all he could tell Will was "Nice hat". The best part is Will looks up at it and grins like a loon.
"I think we can afford to give him one day's head start."
After Elizabeth makes it clear she chooses Will at the end, Norrington tells Will that he expects him to take care of her properly and does it by acknowledging that Will was the one to make the sword, after Will had to watch someone else get the credit for it at the beginning of the movie.
