There are different conversations for different purposes. So I don’t have one favorite conversation; or I might have a favorite, but it might be different if you asked me tomorrow or a week from tomorrow.
I enjoy conversations where you get to learn something important about a character. Sometimes what you learn is funny, sometimes poignant and sometimes creepy as heck.
I’ve always been impressed by the conversation Harry and Michael had over what to do with “Snake Boy” Quintus Cassius, in Death Masks; after Cassius had surrendered his coin, because it showed us Michael’s core beliefs and true character.
Harry - "You've seen these things," I snarled, stalking over to face Michael. "I've seen the corpses they've left. They would have killed me, Susan, you-hell, all of us-without blinking an eye. God only knows what they have in mind with that curse they're putting together."
Michael - "All power has its limits, Harry." He shook his head. "This is the limit of mine."
Harry - "They might already have killed Shiro. And you're going to let this bastard walk?"
Michael - "I know that," he said in that same deadly quiet voice. "I know they've hurt him. That they're going to kill him. Just as Shiro knew that Nicodemus would betray his promise to set you free. It's one of the things that makes us different than they are, Harry. The blood on their hands does not make it right to bloody my own. My choices are measured against my own soul. Not against the stains on theirs." He looked at Cassius, and the Denarian flinched away from the silent flame in Michael's expression. "It is not for me to judge his soul. No matter how much I might want to."
If you didn’t know who Michael was before that conversation, you certainly did after it.
Seeing as the OP asked what our favorite conversations; plural, are, here is one other one.
The conversation Harry had with Murphy in Summer Knight, when Harry finally came clean and revealed the inner workings of the various supernatural powers and conflicts that were raging and Harry’s role in them.
I really felt like the series took a dramatic turn at that point. Because Harry had been pushed to his limits he was finally bringing Murphy fully into the picture. Even though Harry was doing so because at that moment he needed all the help he could get, you knew it would have major ramifications going forward.
Harry - "One last thing. If you come in on this, you have to understand something. You have to promise me that you won't haul SI and the rest of the police in on everything. You can dig up information, use them discretely, but you can't round up a posse and go gunning for demons."
Murphy - Her eyes narrowed. "Why the hell not?"
Harry - "Because bringing mortal authorities into a conflict is the nuclear assault of the supernatural world. No one wants to see it happen, and if they thought you might do it, they'd kill you. Or they'd pull strings higher up and get you fired, or framed for something. They would never allow it to pass. You'd get yourself ruined or hurt or killed and it's likely a lot of people would go down with you." I paused to let the words sink in, then asked, "Still want me to tell you?"
Murphy - She closed her eyes for a moment and then nodded, once. "Hit me."
Harry - "You're sure?"
Murphy - "Yeah."
Harry - "All right," I said. And I told Murphy all of it. It took a while. I told her about Justin and about Elaine. I told her about the supernatural forces and politics at play in and around the city. I told her about the war I'd started because of what the Red Court had done to Susan. I told her about the faeries and Reuel's murder.
And most of all, I told her about the White Council.
Murphy - "Those spineless, arrogant, egomaniacal sons of bitches," Murphy growled. "Who the hell do they think they are, selling out their own people like that?"
That was the moment when Karin Murphy went from a sometime antagonist to full ally and it was glorious.
Maybe there should be a separate thread for what are our favorite joke lines and conversations in the Dresden Files, because there are certainly a good number of those.