Catelyn Stark is one of the few voices of reason in A Song of Ice and Fire, but she is repeatedly ignored, because she is a mother, and because she is a woman. Although, like every other character in the series, she is fallible (trusting Littlefinger, for example), her advice is generally sound. She has lived through one bloody war and lost people she cared about as a result. She does not want to live through another one and lose all her family. And so she alone, out of all the main characters, speaks out against war and vengeance. She understands that further death and destruction will not bring back the people they have lost. More than anything, she wants peace. She warns Renly that his men are the "knights of summer," playing at war with no understanding of its reality. She combines experience with wisdom, and many of the terrible situations in the books could have been avoided if people listened to warnings.
But they don't. Because she is a woman, and because she is a mother. Because, as a mother, she is dismissed as too soft-hearted, too concerned with protecting her children to understand the true nature of war. Because, as Robb's mother, heeding her words would be seen as weakness.
Some readers criticize Catelyn for involving herself in the war instead of returning to Winterfell to care for Bran and the increasingly feral Rickon. They argue that Catelyn should have hurried back home as soon as she learned that Bran was awake. They blame the disaster at Winterfell on Catelyn, even though Catelyn explicitly warned Robb against sending Theon as his envoy, because she should have been present to nurture and protect her children.
Except Catelyn did dedicate herself entirely to Bran, for two weeks. She sat by his bedside, refusing to eat or sleep, completely lost in grief as she willed him to recover. And her presence saved his life from the assassin. But the assassin attack also wakes Catelyn up from her grief and makes her realize that she cannot help Bran by grieving at his bedside. She must warn Ned about the attack, and later, she must help Robb succeed in his war. As far as Catelyn knows, Bran and Rickon are safe in Winterfell under the care of Maester Luwin. They will not be attacked or put at risk unless Robb loses the war. And so her biggest concern is offering Robb, the young king with no experience of leadership or war, her advice and support. By staying by Robb's side, by working as envoy to Renly, she is actively protecting all of her family. And through her efforts to end the war, she is actively trying to protect everyone she encounters.
Catelyn is also despised for her treatment of Jon, for telling him that he should have fallen from the tower instead of Bran. It's a shocking thing to say, but Catelyn is lost in grief for her son at the time. As a general rule, she is not cruel to Jon. She allows him to live in Winterfell, allows him to form close friendships with her own children, and generally allows him to be treated like any of the Starks in the castle. She resents him, because he is living, breathing proof of her husband's infidelity, walking around right under her nose, looking more like her husband than any of her own children, but she does not treat him with anything worse than indifference, except after two weeks of sleeplessness and grief.
The strongest and strangest criticism of Catelyn, however, is what she does after she leaves Bran and travels to King's Landing. Although the above instances criticize Catelyn for not being enough of a caring mother, Catelyn's dealings with Tyrion Lannister are criticized because Catelyn is too much of a mother, putting concern for her family above reason. Despite the fact that one of the series' major themes is that all events are interconnected, and no single action or individual is completely responsible for anything as complicated as a war, many people claim that Catelyn is responsible for the start of the war because she believes Littlefinger's lies and captured Tyrion.
But Catelyn does not choose to capture Tyrion. She aims to return to Winterfell unnoticed and attempts to hide from him when he enters the inn where she is staying. But once he does spot her, she has no choice but to take action. She believes that Tyrion is part of a plot against her family, and that if the Lannisters find out she visited King's Landing, her husband and children will be at risk. What other choice does she have? She takes him to the Eyrie because she believes that her sister is trustworthy and has good reasons for suspecting the Lannisters of murder. Once she finds out how unhinged Lysa has become, she attempts to act as the voice of reason and protect Tyrion from Lysa's bloodthirsty reaction, but it is too late for her to make a difference. Catelyn's actions do prompt Jaime Lannister to attack Ned in the street and for Lannister men to start attacking the Riverlands. But the Lannisters were already spoiling for a fight. Lysa Arryn is partly responsible for lying to her sister. Littlefinger is definitely responsible by making Catelyn believe Tyrion attempted to murder her son. It is a complicated web, and Catelyn is only one imperfect actor inside it.
Finally, Catelyn is often criticized for releasing Jaime Lannister, again putting her motherly concerns over the practical considerations of war. Diplomatically and strategically, it doesn't appear to be the best move (although, as the plot progresses, it actually turns out to be the best thing she could have done). But Catelyn was grieving for her children. Jaime's release is not the move of a cold, calculating leader. But it is the act of a relatable human being, and I, at least, admire her for it.
As Brienne says, Catelyn has "a woman's courage," and a woman's strength. As well as her intelligence and insight, Catelyn's biggest strength is her ability to endure. Over the course of two books, she loses almost everything. Yet she keeps going. She keeps insisting that vengeance is misguided, that they should all hold on to what they have left, before nothing remains at all. And people keep ignoring her, because she is a mother, because she is too motivated by soft feminine emotions, and because she should stop invading in men's space and return to her children. And so she stands relatively powerless, like Cassandra, warning everyone of their mistakes and watching, helpless, as they ignore her and she loses everything loves.
And all the while, readers criticize her for her mistakes. For being both too motherly and not motherly enough. For involving herself in events, and for daring to be a full and complex character, instead of simply a "mother."