Families in A Storm of Swords are extremely large and tight-knit, and they provide their members with guidance, support, identity, status, and power. Family allegiance, therefore, is of vital importance, and different families that share close ties often function like a single, extended clan. The various families of the North, for example, generally act as a united front against the major families of the South, despite whatever personal struggles may be occurring between them. These family alliances have a great deal of power, and so social contracts like marriage that formally join families together become critically important in the politics of Westeros. King Joffrey's betrothals illustrate this point. He was initially engaged to marry Sansa Stark as a way to unite their families, but when it became clear that the union afforded no political advantage, Joffrey was quickly engaged to Margaery Tyrell instead. When these same social ties are broken, the results can be what is fundamentally clan warfare. Robb Stark's decision to marry Jeyne Westerling rather than a woman from the Frey family, as he promised, provoked the Freys into a massacre of the Starks and their allies at the Red Wedding.
The world we see in the novel is mostly male dominated. Male monarchs rule the lands, male advisors counsel them, male knights fight their wars, and women are married off in order to secure lands and titles. Male characters constantly threaten women with abuse, and many women are treated as property or sexual objects. (Notably, this rubric is consistent with European gender dynamics during the Middle Ages.) But the novel’s major female characters generally defy these conventional roles. Daenerys, for instance, is a strong-willed leader. Cersei wields significant power in King's Landing and is at times the de facto ruler of Westeros. But no characters chafe against conventional female roles as Arya and Brienne do. From the first, Arya has been more interested in swordfighting than needle work, which is why naming her sword Needle was loaded with irony, and she bristles when anyone suggests that she should behave a certain way because she's a girl. Brienne similarly eschews feminine trappings and refuses to be dominated by any man. She may also be the only female knight in existence, and despite the verbal and sometimes physical abuse she suffers for her dress and behavior, she continues to behave as she wants, not as others think she should.
There are instances where major female characters play traditional female roles, but even in those instances they turn these roles to their advantage. Cersei, notably, feels she must fulfill a conventional female role, even as laments at times that she wasn't born a man. She thinks she must always be beautiful, and she doesn't think she would ever be able to rule Westeros openly. But she also uses men's perceptions of her as a sex object to her advantage, often manipulating them into doing what she wants in exchange for sexual favors, or even just the hope of them. Catelyn Stark hews to a conventional female role as well, albeit in a much different way. She is a typical mother figure, and much of what she does in the novel she does in order to take care of her children, particularly Robb. That need to protect her children is the source of a great deal of her strength.
As the war progresses, the royal houses keep track of crimes and betrayals, and for each perceived misdeed, a character often seeks revenge. The need for revenge is so prevalent in the novel that it motivates a great deal of the novel's plot and serves as a driving force for several characters. It also leads to the deaths of numerous characters. In some instances, revenge is sought for relatively minor infractions like an insult to a character's or family's honor. When Robb breaks his engagement to a Frey daughter, for instance, the Freys get their revenge by assassinating Robb and Catelyn Stark at the Red Wedding. In other instances it stems from long-standing grievances, as when Tyrion vows vengeance against Jaime and then murders his father in revenge for their mistreatment of him and Tysha, the woman he once loved.
In many other cases, characters seek retribution for the murder of their friends or family members. A reanimated Catelyn Stark joins the Brotherhood Without Banners to exact revenge on the Freys for their massacre of her family at the Red Wedding. Arya lives obsessed with gaining vengeance on those who have hurt or killed her family and friends, and she regularly recites a list of names to herself of those people so she won't forget them. Tyrion's champion at his trial, Oberyn Martell, takes up the role so that he can get revenge against Gregor Clegane, the other combatant, for violence Gregor committed against Martell's family in the past. Occasionally in these situations, the revenge exacted by one character causes another character to want retribution, perpetuating the need for revenge.
Several characters endure struggles with their identities in the novel, as events force them to reconsider how they think of themselves and what person they want to be. Jon Snow faces two such struggles. First, as he pretends to be a wildling, he finds himself fitting in among the wildlings more than he anticipated, and he even breaks his vows to the Night's Watch by getting involved in a romantic relationship with Ygritte. Ultimately he must decide whether he wants to become one of them or return to the Watch, and he struggles a great deal with the decision. Later, he is offered the chance to rule over Winterfell by Stannis but would have to leave the Watch to do so. This situation taps directly into the identity issues Jon had growing up, when he felt himself not fully a part of the Stark family, and he feels torn again between what are essentially two versions of himself.