JACK: You don't want to do this.
Jack and Rose meet when Rose is contemplating throwing herself off the back of the Titanic. Jack's not convinced that Rose genuinely wants to die, and manages to get her to calm down/try to get back over the rail.
JACK: Listen to me, I've got you. I won't let go. Now pull yourself up, come on!
"Not letting go" is a big theme in the movie—you'll see that phrase come up again and again. In this first instance, Jack is trying to get Rose back over the railing…but she slips. Luckily for Rose, Jack remains very committed to keeping Rose alive throughout the movie.
JACK: No, no, that's not what I was thinking. What I was thinking was, "What could have happened to this girl to make her think she had no way out?"
When Jack and Rose talk later, Rose is worried that Jack thinks badly of her, given that she's clearly so privileged and, yet, was trying to kill herself. Jack explains he's not the judgmental type—really, he was just interested in knowing why she was so distraught.
JACK: I don't think so. They've got you trapped, Rose, and you're going to die if you don't break free. Maybe not right away because you're strong, but sooner or later, that fire that I love about you, Rose…that fire's going to burn out.
ROSE: It's not up to you to save me, Jack.
JACK: You're right. Only you can do that.
Having explained earlier that she's basically miserable and trapped in her relationship with Cal, Rose is now trying to convince Jack that it's all good—and that he needs to leave her alone. He isn't having any of it, since he thinks that Rose's current life will eventually kill her, one way or another.
JACK: Don't you do that. Don't you say your goodbyes. Not yet.
At the end of the film, Jack and Rose are literally chilling in the water, the boat having sunk, and Rose tells Jack that she loves him. That's a nice sentiment, but it makes Jack nervous, since he takes it as Rose saying her goodbyes.
JACK: You must do me this honor. You must promise me that you'll survive. That you won't give up, no matter what happens, no matter how hopeless. Promise me now, Rose, and never let go of that promise.
Before Rose can get too down in the dumps (she's freezing in the Atlantic, after all) Jack makes her promise that she's going to keep fighting her way through their current predicament—and any others that might arise.
FIFTH OFFICER LOWE: We waited too long.
When one of the crewmembers brings an empty lifeboat back to look for survivors, he realizes that most everyone has died, and that they failed to do their duty to save/help the other survivors.
ROSE: I'll never let go. I promise.
Unfortunately, Jack doesn't make it through the night, but Rose makes good on her promise to keep on keeping on, reaffirming that vow as she releases Jack's body into the sea.
ROSE: Her husband had struck gold someplace out west, and she was what mother called "new money."
This is Rose describing Molly Brown. Molly seems like good people to us, but people like Rose's mother are always quick to distinguish themselves from "new money." Plus, Ruth thinks Molly is vulgar to boot…
TOMMY: Ah, forget it, boyo. You'd as like have angels fly out of your arse as get next to the likes of her.
When Tommy sees Jack checking out Rose, he tries to quash any hopes he might have of getting with the young beauty. Since Rose is clearly of a different class, Tommy doesn't think Jack has a chance.
CAL: What made you think you could put your hands on my fiancée?! Look at me, you filth! What did you think you were doing?
When the ship's crew mistakenly thinks Jack had been attacking Rose (when he was actually trying to help her), it's understandable that Cal is upset… but he definitely shows his snobby stripes in the language he uses to communicate his anger.
CAL: Well, it's for royalty. We are royalty, Rose.
Cal gives Rose a huge-tastic diamond necklace called the Heart of the Ocean. He senses that she hasn't been feeling all that chipper, and so he decides to give her something really expensive and elite…since, you know, that's the kind of thing that would cheer him up.
UNNAMED COUNTESS: Look, here comes that vulgar Brown woman.
RUTH: Quickly, get up before she sits with us.
Remember how we said Ruth didn't like Molly? Well, this is the kind of thing we were talking about. Ruth is trying to escape from the lunch table before Molly can join her and her friends. Ruth does not consider Molly to be on her level in the least, and she barely hides her disdain.
MOLLY: Well, you're about to go into the snake pit. What are you planning to wear?
It seems that Molly doesn't think that highly of Ruth and her friends, either, since they are the members of the "snake pit" she mentions here when talking to Jack. You see, Jack is supposed to come to dinner with Rose, Cal, etc., as a thank you for saving Rose's life…and as Molly points out, he'll probably need something posh to wear.
CAL: Dawson! That's amazing! You could almost pass for a gentlemen!
Cal manages to insult Jack while complimenting him, of course. Sure, he seems impressed at how well Jack cleans up…but still says that Jack can only "almost" pass for a gentleman.
ROSE: They assumed he was one of them—heir to a railroad fortune, perhaps. New money, obviously, but still a member of the club.
In voiceover, Rose describes how the posh types reacted to Jack's appearance among them. Apparently, he passed quite nicely for a gentleman (Cal's little joke aside).
RUTH: Tell us of the accommodations in steerage, Mr. Dawson. I hear they're quite good on this ship.
In an effort to remind Jack of his real "place," no doubt, Ruth's first question is about how he enjoys steerage. "Classy."
ROSE: Is that the going rate for saving the woman you love?
Rose isn't impressed with Cal's attempt to thank Jack for saving her life by just tossing money at him. For Cal, of course, love is pretty transactional, so money is the perfect way to show love.
CAL: You know, there's nothing I couldn't give you. There's nothing I'd deny you…if you would not deny me. Oh, open your heart to me, Rose.
Again, Cal is talking about matters of the "heart," but he's making it all about the stuff he could give her—not feelings.
JACK: Do you love him?
ROSE: Pardon me?
JACK: Do you love him?
ROSE: You're being very rude. You shouldn't be asking me this.
Jack asks what he thinks is a natural question, after hearing about Rose's nerves regarding marriage. However, it strikes a nerve with Rose.
ROSE: I think you must have had a love affair with her.
Rose is assuming that Jack was romantically involved with one of his models because she appears more than once. However, it turns out that Jack just liked drawing her hands, and they had no romantic involvement at all. Jack is a lot more nuanced than Rose assumes.
ROSE: My heart was pounding the whole time. It was the most erotic moment of my life. Up till then at least.
Here, Rose is describing when she posed nude for Jack while wearing the Heart of the Ocean—Cal's token of his own "love." She and Jack haven't formally cemented their own love with words, but this moment is clearly about Rose taking back her own heart and thumbing her nose at Cal. By commissioning the drawing and then leaving it for Cal with a note, she's basically throwing her disdain for him (and his heart—diamond or otherwise) in his face.
ROSE: I love you, Jack.
Toward the end of the movie, Rose finally utters those three little words that every boy freezing to death in the ocean wants to hear: I love you. Unfortunately, Jack actually doesn't want to hear it, since he believes this is Rose saying her goodbyes.
ROSE: It was the ship of dreams to everyone else. To me, it was a slave ship, taking me back to America in chains.
Obviously, this is a pretty over-the-top claim, since she's not being physically confined, whipped, or starved. However, it does speak to how completely trapped and helpless she felt to get out of her engagement to Cal.
ROSE: It was everything. It was my whole world and all the people in it. And the inertia of my life. Plunging ahead and me powerless to stop it.
Rose is trying to explain to Jack why she was miserable enough to consider suicide. Apparently, she feels like she's just kind of being propelled toward a fate she doesn't like without any power to stop herself.
ROSE: I see you had that undertaker of a manservant follow me. How typical.
Cal is fond of having his valet, Lovejoy, follow Rose. Naturally, that kind of thing only contributes to her sense of being trapped.
CAL: My fiancée! Yes, you are! And my wife in practice, if not yet by law, so you will honor me. You will honor me the way a wife is required to honor a husband. Because I will not be made out a fool, Rose. Is this in any way unclear?
When Cal finds out that his prize possession, Rose, has been out frolicking below deck with Jack, he is not pleased. He seems to take it as infidelity, so he throws a huge tantrum with Rose at the breakfast table. When he says that Rose will "honor" him, he basically means obey. Bottom line: Rose's attempt to break free got her in serious trouble.
ROSE: How can you put this on my shoulders?
Ruth is trying to convince Rose not to see Jack anymore, and she's resisting. Rose doesn't understand why the entire future of the family depends on her marital decisions. It all falls on deaf ears, though, since Ruth basically tells her to suck it up.
ROSE: Mr. Andrews, thank God! Where would the Master at Arms take someone under arrest?
Jack gets literally imprisoned right around the time Titanic hits the iceberg. He's handcuffed to a pipe below deck, as water is rushing in.
ROSE: There's no key.
Rose goes to rescue Jack from where he was being detained, since he's certain to drown otherwise, but can't find the right key to the handcuffs that link Jack to a pipe. So, she gets an axe and frees him that way.
TOMMY: For God's sakes, man, there are women and children down here! Let us up so we can have a chance!
In an effort to keep the lifeboat situation under control (and probably to save room for the first-class passengers), the crew had locked the gates from the lower decks to the upper. Tommy Ryan and the others are furious, of course, since locking them below deck is basically condemning them to drowning.