Lords of the Riverlands from their fortress of Riverrun. Unlike many of the other Great Houses, they were never Kings of their own realm, but were granted lordship under Aegon the Conqueror. Their current head, Hoster Tully, has three children: the two daughters Catelyn and Lysa, who married Eddard Stark and Jon Arryn, and the son Edmure, who will succeed him when he dies. They ally with the North during the War of the Five Kings. Their words are "Family, Duty, Honor" Their members tend to have fiery red hair and their sigil is a leaping silver trout.
House Tully is an old noble house, dating back to the Age of Heroes. Unlike many Great Houses they never ruled as kings, but held Riverrun for a thousand years as vassals of those who did. Around the time of Aegon's Landing, the Riverlands were under control of Ironborn led by House Hoare. Their king, Harren the Black, had forced the river folk to build a massive fortress know as Harrehall and completely draining their resources dry. Hoare was hated among the Riverlords and when Aegon Targaryen started his campaign of conquest, they promptly deserted the Iron King, and led by Lord Edmyn Tully, they had sided with the Targaryens during the War of Conquest. As such, the Tullys were rewarded as Lords Paramounts of the Trident.
Lord Hoster Tully has took a ward, Petyr Baelish, then an obscure lordling from the Fingers. Petyr fell in love with Lord Tully's eldest daughter Catelyn, but was rebuffed. When Hoster finally accepted an offer from Lord Rickard Stark that Catelyn be betrothed to Brandon Stark, heir to Winterfell. Petyr Baelish dueled Brandon for Catelyn's favor, resulting in his exile from Riverrun. Brandon and Rickard's however were executed by the Mad King and the Tullys joined the the ensuing rebellion against him, arranging for Catelyn to wed Brandon's brother Lord Eddard and for her sister Lysa to marry Lord Jon Arryn. The Riverlands was a major theater of the war, and Lord Hoster fought in the Battle of the Bells and crushed the royalist houses of Goodbrook, Darry, Mooton, and Ryger. His control over his bannermen was not complete; aside from these recalcitrant houses, House Frey delayed their contribution until after the decisive Battle of the Trident, itself fought in the Riverlands. After the war, the Tully girls rode off to their husbands' seats.
Due to their geographic position, their central, flat location ensures that the Tullys will be drawn into the fight whenever there is a war between the other houses, whether they like it or not. As such, the Riverlands serve as main battleground during the War of the Five Kings and is the region that's hit most hard during the conflict.