DON CORLEONE: I'm gonna make him an offer he can't refuse.
Here it is—the classic line. Part of its inherent irony is that while an "offer you can't refuse" sounds like it might be so amazing you'd be silly to refuse it, in this case, it probably means holding a gun to your head and forcing you to do something.
MICHAEL: My father is no different than any powerful man, any man with power, like a president or senator. KAY: Do you know how naive you sound, Michael? Presidents and senators don't have men killed. MICHAEL: Oh. Who's being naive, Kay?
Michael takes a pretty cynical view of politics. In his view, a politician who starts a war or causes the loss of life for personal gain or impure motives is no different than a mafia boss… Maybe he has a point? Sounds kind of like House of Cards.
TOM HAGEN: The Senator called, he apologized for not coming but said that you would understand; also some of the judges. They've all sent gifts.
Highlighting Michael's point from Quote #2, Don Corleone has friends who are politicians and are in cahoots with his criminal enterprise. Success seems to correlate with dishonesty and treachery.
CLEMENZA: Leave the gun. Take the cannoli.
Not exactly a profound quote, but maybe intelligent advice: In this case, the gun, presumably free of fingerprints, is left behind after killing someone, while the delicious cannoli is taken along. Just because you're a vicious murderer doesn't mean you need to abandon your appetite, apparently.
SOLLOZZO: I don't like violence, Tom. I'm a businessman; blood is a big expense.
Sollozzo highlights the fact that there's nothing actively malicious in the violence he commits, so you shouldn't take it personally. It's just a means he uses to get his job done, and a messy means at that.
MICHAEL: Where does it say that you can't kill a cop?
Michael is, to cite a cliché, "thinking outside the box." He wants to wipe out the corrupt cop Captain McCluskey because the Corleones are just too far down in the count. Yet directly attacking an authority figure is more dangerous than killing other gangsters because it poses a greater threat to society.
MICHAEL: It's not personal, Sonny. It's strictly business.
Sonny accuses Michael of taking things personally, but, in reality, he is the hothead. Michael is actually the cool and collected one who can think ahead and strategize.
SOLLOZZO (in Italian): I am sorry. What happened to your father was business. I have much respect for your father. But your father, his thinking is old-fashioned. You must understand why I had to do that. Now let's work through where we go from here.
Sollozzo is trying to explain that he doesn't hate Vito Corleone, so Michael shouldn't overreact to the assassination attempt. It was just business, just a way of getting something that he couldn't get through negotiations. But it's too late for that excuse—Sollozzo's simply gone way too far.
CONNIE: Papa never talked about business in front of the kids.
Connie believes that family life and mob business are two separate spheres. If you're a mob boss, you need to live a kind of contradiction: playing with your kids in one world, ordering people's deaths in the other.
MICHAEL: Don't ask me about my business, Kay.
At the end of the movie, Michael refuses to answer Kay's question at first (as to whether he had Carlo killed or not). But she insists, so he lies and says he's not responsible. He's trying to live the contradiction mentioned in Quote #9: family man in one world, ruthless killer in the other.