In this chapter, Joab convinces David to bring Absalom back from exile.
Taken as a whole, I think this chapter also shows David to be a largely passive agent, as other people move and act around him. Other people come and ask him for things, but David does not seem to be taking the initiative I would expect in this story. It's possible that David is reluctant to punish Absalom when David himself committed murder by killing Uriah. This unfortunately shows us how sin in the highest reaches of government can ripple outward into society, as David's sin opens the door for many others to commit the same crimes that David himself committed.
First of all, David's "heart is inclined towards Absalom", and he "mourned for his son every day". But David doesn't take any action towards this, and it's only when Joab prods him that David allows Absalom to return.
Joab hires a "wise woman" to go and convince David to reconcile with Absalom. This story is strangely reminiscent of Nathan's parable through which he rebuked David in 2 Samuel 12, which makes me wonder if the "wise woman" is some sort of term for a female prophet. The banter between David and the "wise woman" is amusing, but I don't see anything there that requires much explanation.
Verse 7 says, "and destroy the heir also", by which the woman's (metaphorical) family gives away their true intentions. Although ostensibly they are killing the woman's son because he committed murder, she is implying that her extended family wants to kill him so that the woman can be left without an heir, and the man's inheritance would go to them instead. The woman, for her part, depends on her sons to provide for her old age, and of course she wants her husband and her family to continue in the land through her remaining son. Of course, none of this actually happened, but it's what she is trying to imply as her family's motivation.
In verse 11, the woman asks David to "remember the LORD your God", by which she is asking him to swear an oath confirming his judgment on her behalf. When David says, "as the LORD lives", that is a ritual formula indicating that he swore an oath in the name of the LORD to do what she asked.
David apparently perceives that Joab is behind this woman's behavior, which makes me wonder two things. First, how did David know that Joab was behind her, and secondly, why does Joab care about David's reconciliation with Absalom? We can see that Joab perceives something in David, that David wanted to be reconciled with Absalom. Perhaps Joab was concerned that this was becoming a distraction to David, or possibly he was seeking to ingratiate himself with Absalom. Joab remains loyal to David, but it's possible that he's trying to build alliances with "the princes", the future rulers of Israel. I should mention, Absalom is not David's heir; David's second-born (after Amnon, who died) is Kileab. Absalom is David's third-born, so it would only be with Kileab's death that Absalom would reign. Besides this, Kileab is a largely inconsequential figure, as David eventually declares Solomon to be his successor.
So I'm not sure. I don't understand Joab's motivation here, but what is clear is Joab's influence. Joab convinces David to bring Absalom back, and then convinces David to allow Absalom to meet him again. This second part is significant. In verse 24, we are told that David allows Absalom to return, but "let him not see my face." What this means is that David is allowing Absalom to live in his own home with his family, but he is denied having any political influence and David shows his disapproval in this way.
Absalom his to burn Joab's fields to get Joab to see him (Joab is possibly avoiding Absalom because Absalom is in disfavor with the king). Joab goes to talk to David, and David allows Absalom to see him, which basically means that he is politically rehabilitated as David has ultimately decided to not punish Absalom.
What I see here is Joab acting as a major power broker, and Absalom learning that he can murder people with impunity because David refuses to punish him.
There are a few more interesting details in this chapter. One is that Absalom names his daughter after his sister, Tamar. Secondly (and more important to the story) is that Absalom is "highly praised" and very handsome. We discover that he also has abundant quantities of hair, which is a symbol in the bible for male strength, youth and fertility. In this respect, Absalom is most reminiscent of Saul, who was also a tall, handsome and strong figure, but a terrible king. Absalom's beauty will play a role in later parts of the story, as he will cast his aspirations for the throne in a little while.