Riverrun and its people don't play too large of a role in A Clash of Kings, but we are introduced to several key members of Catelyn's family here. Here's the Riverrun rundown:
Edmure Tully is the acting lord of Riverrun and Catelyn's brother. Although well-intentioned, Edmure is soft-hearted and not much with war tactics—his lack of tactical forethought is clear when he asks Robb to let his riverlords defend their own lands rather than consolidating their strength (8.Catelyn.45), and he also opens his castle to frightful peasants despite a likely siege to come (40.Catelyn.100-101).
Despite his strategic shortcoming, Edmure still manages to defeat the Mountain's forces at the Battle of the Fords, showing courage, and perhaps even a growing aptitude for war.
Hoster Tully is Catelyn and Edmure's father and the lord of Riverrun. On his death bed, Hoster has grown frail and senile, so his mutterings make little sense to anybody (8.Catelyn.83)
Brynden is brother to Lord Hoster Tully, and Catelyn's uncle. A precise and realistic man, Brynden serves as Robb's scout and understands the cost the war will have on both sides that fight (8.Catelyn.99). He is called "the Blackfish" (8.Catelyn.91) because Hoster once disowned his brother. The reasons for their sibling squabbles are unknown right now.
A bard hanging out around Riverrun, he writes songs telling of Robb and Edmure's glorious victories on the battlefield.