After the whole Kissing Scene, Pavel casually challenges Bazarov to a duel.
No big thing.
He also proposes that they use pistols.
Pistols = Certain Death.

Bazarov agrees. He knows that he has got himself into this situation and must accept the challenge.
There is an understanding between the two that there will be no records of their duel. No witnesses, no statements, no papers. If there were records or questions asked, then Fenichka's disloyalty would be revealed and Nikolai's heart would be broken.

Pavel putting his life on the line to uphold his brothers honor is just another example of his Romantic disposition. Challenging someone to a duel to protect honor is such a classic, old way of settling issues. The fact that it is a secret duel makes it all the more Romantic. He is doing it to regain his brother's honor, though, without anyone knowing. It displays a personal, empathetic sentiment.

The two meet in the forest later the next day, both bearing pistols. The parameters of the duel are set and the two take their strides, turn, and

SHOOT!

Pavel's bullet just barely misses Bazarov's ear, piercing a tree directly behind him instead, and Bazarov manages to shoot Pavel in the leg.
Pavel is injured but conscious, and offers his life up to Bazarov, reminding him that part of the deal was to have two bullets. Bazarov internally smiles at Pavel's chivalry and organizes a way to get him home, and it becomes Bazarov's duty to actually tend to Pavel until Nikolai can retrieve a doctor. (Nikolai is led to believe the whole thing was an accident).
Pavel becomes somewhat delirious and overly happy. I think he experiences a sort of high off the adrenaline, and he even seems to have forgotten all animosity toward Bazarov, "Nikolai Petrovich kept stealthily wringing his hands, while Pavel Petrovich laughed and joked, especially with Bazarov" (128).

It is while Pavel is still on his Adrenaline High that he notices Fenichka, "Pavel Petrovich gazed at her intently, and drank off the glass to the last drop" (129). Readers believe that he is looking at her intently with anger or suspicion because of her infidelity, but also because that is how he always related to her, that is until he says, "Don't you think, Nikolai, Fenichka has something in common with Nellie?" and by this he means, "Princess R-. Especially in the upper part of the face" (129).

" 'Ah, how I love that light-headed creature!' moaned Pavel Petrovich, clasping his hands mournfully behind his head. 'I can't bear any insolent upstart to dare to touch...' he shispered a few minutes later.
Nikolai Petrovich only sighed; he did not even suspect to whom these words referred"

PAVEL LOVES FENICHKA! She has a similar appearance to his true love the Princess R- who deserted him earlier in his life! This whole time readers thought that he would gaze in at her to make her feel uncomfortable and to check in on her, but really it was because he loved her and wanted to see her. Fenichka was always afraid of Pavel because he was always staring at her with a "knowing look," like he ws judging her. Now we know that he stares at her because he longs for her.

When he says, "I can't bear any insolent upstart to dare to touch..." he means Bazarov because he just witness him kissing Fenichka, but could he also mean Nikolai? His own brother? He must have some jealousy in the matter. Nikolai doesn't even realize it, even when Pavel made the blatant comment about Fenichka and Princess R-'s resemblance.

Any jealousy or selfish thoughts that Pavel might have had are overruled by his love for his brother. When he gets Fenichka alone, he pushes her to stay true to only Nikolai.

"Fenichka!" he was saying in a strange whisper; "love him, love my brother! He is such a kind, good man! Don't give him up for any one in the world' don't listen to any one else! Think what can be more terrible than to love and not be loved! Never leave my poor Nikolai!" (132)

He ends in convincing Nikolai to must marry Fenichka already. It is the right thing to do. Who cares about her status or the fact she got pregnant out of wedlock in a not-so-proper way. Just marry her! Pavel knows how it feels to love someone and then get rejected. He does not want his brother to experience that as well, so he gives up any chance he might have had with Fenichka to protect his brother.

The last paragraph in the chapter struck me as rather Romantic of Turgenev and a good example of the author's beliefs in Romanticism.

Pavel Petrovich moistened his forehead with eau de cologne, and closed his eyes. His handsome, emaciated head, the glaring daylight shining full upon it, lay on the white pillow like the head of a dead man... And indeed he was a dead man. (134)

So either Pavel died or Turgenev is making a statement about how Pavel was a dead man, a dead man in the case that he was alone, had loved and had not been loved in return, and because of this, his life had no meaning = dead man.

Cut to Odinstovas:
Arkady and Katya are a lot alike:
Both are in a way "subordinate" to either Odinstova (Katya to her sister), or Bazarov (Arkady to his mentor). Arkady and Katya are sitting outside of Odinstova's home chatting. Arkady reveals that he holds Katya's presence and her opinion higher than anyone else's, even over Odinstova's, which Katya didn't expect.
Katya is honest with Arkady about her opinions on Bazarov. She explains that Bazarov and him are two very different people. She says that Arkady shouldn't try to be like him. Bazarov is wild and fleeting and restless, while Arkady is tamed.

 The whole scene is very peaceful and watching the two, it is apparent that they are very comfortable with each other and respect one another. Arkady seems to be completely relaxed and himself when he is with Katya. This is apparent when his demeanor completely changes when Bazarov arrives at Odinstova's home. Arkady is dismayed and there is a very obvious change in his emotions and attitude.
Odinstova comes and talks to Bazarov in a very formal way. They basically act as if they never had feelings for one another. What bugs me is the two talk about Arkady and Katya as if they are significantly younger than they are, when in reality, they are only 3-4 years their junior.

"By the way, did you know I used not quite to understand your close friendship with Arkady Nikolaich; I thought him rather insignificant. But now I have come to know him better, and to see that he is clever... And he's young, he's young... that's the great thing... not like you and me, Evengy Vassilyich (Bazarov)" (142). 

The thing is, they make themselves old. If they wanted to be young and foolish, they could be. If they could just let down their guard and allow themselves to express their true feelings toward one another, they could live in innocent, immature, ignorant bliss as well. I personally think it is their own fault that they are not happy.

Anyway, Arkady realizes that he does not look up to Bazarov any more, his admiration for him has evaporated, and he is also no longer jealous of Odinstova's obvious preference for Bazarov over himself. This shows a growth in character on Arkady's part. He has matured, but it has somehow made him appear more free and young, most people link 'mature' with old, sad, aged, but Arkady seems freed by it. Before, he was juvenile and blindly following Bazarov, which made him look childish, but it also made him tired, going against his true beliefs, and old. He was shackled down by trying so hard to be a strict Nihilist. Now he is free from being herded along by Bazarov. Though Arkady's annoying despondence that was very child like is gone, his maturity makes him free, and that freedom makes him young. Arkady recognizes the change within himself, attributes it to the lessons he learned from Katya, and he proposes to her.

There is another proposal that also takes place. Odinstova reaches out to Bazarov, asking him to stay, but Bazarov claims that he has been "moving too long in a sphere which is not [his] own" (147). I think Bazarov is tired out by the company he has been keeping. He is exhausted by all the attempts he has made at women and how they have all failed. I also think he doesn't want to be a charity case for Odinstova. I think he looked at her invitation to stay as a way of smoothing over the ice from what had happened earlier in their relations. "Smoothing out" being the key thing, not offering to make any further advancements. Bazarov does not want that. He either wants Odinstova as a lover or not at all. He bids adieu and leaves, knowing he will probably never see Katya, Arkady, or Odinstova ever again.