In revenge for the imprisonment of Tyrion, Tywin attacked the Riverlands, the place of birth of Catelyn Tully (the Tullys are the Lords of Riverrun and all the Riverlands, like the Starks with Winterfell and the North). In the Riverlands the army of Jaime defeated the Tully army of Edmure Tully (the brother of Catelyn and Lysa) and besieged Riverrun.
At the same time Tywin gathered his forces with the troops of Gregor Clegane (who defeated the troopers sent by Eddard to capture him) and took Harrenhal, an ancient and ruinous fortress in the Riverlands (and the biggest castle in the Seven Kingdoms). Then he marched to attack the Northern army but Robb tricked him and Tywin only fought against a diversion army commanded by Roose Bolton (a Stark bannerman and Lord of Dreadfort). Roose Bolton was a proven commander and he could retreat his troops when he lost the battle, avoiding a greater loss of troops.
Despite this, Lord Halys Hornwood was killed, Lord Medger Cerwyn, Ser Wylis Manderly and Harrion Karstark and four Freys were all captured (Cerwyns, Hornwood Karstark, and Manderly are some of the noble houses of the North.) Robb could free Riverrun and capture Jaime.