Rickard was the Lord of Winterfell, the Warden of the North and head of House Stark until he was executed by the Mad King, Aerys II Targaryen. He was the father of Brandon, Eddard, Lyanna and Benjen.
Rickard Stark was the head of House Stark and Lord Paramount of the North. The North is one of the constituent regions of the Seven Kingdoms and House Stark is one of the Great Houses of the realm. House Stark rule the region from their seat of Winterfell and Rickard also held the title Lord of Winterfell. He was also the Warden of the North.
He served under King Aerys II (known as the Mad King). He is the father of Brandon, Eddard, Lyanna and Benjen Stark. He arranged for the betrothal of his daughter Lyanna Stark to Robert Baratheon, the Lord Paramount of the Stormlands. He also betrothed his son and heir Brandon Stark to Catelyn Tully, the daughter of Hoster Tully, Lord Paramount of the Riverlands. Both engagements were broken when Prince Rhaegar Targaryen abducted Lyanna. Brandon protested Rhaegar's actions and King Aerys responded by summoning Rickard to court in King's Landing. Both Brandon and Rickard were killed by the Mad King. These events provoked the civil war known as Robert's Rebellion.
When Lyanna was kidnapped by Prince Rhaegar Targaryen, the son of the King, Rickard's eldest son and heir, Brandon, rode to King's Landing seeking justice. Aerys II imprisoned Brandon and ordered that Rickard ride south to answer for his son's impertinence. Against advice, Rickard agreed, assuming that the matter would be settled through a trial or a bargain of some kind. Instead, Aerys had Lord Rickard burned alive while Brandon watched, and then had Brandon strangled.
Aerys then ordered Jon Arryn to hand over his ward and new Lord of Winterfell Eddard Stark. Having sworn a sacred oath to protect Eddard, Jon instead rose his banners in rebellion. He was supported by the North, Lyanna's betrothed Robert Baratheon, and Hoster Tully. This sparked the beginning of the war known as Robert's Rebellion, which led to the near-destruction of House Targaryen and its removal from the Iron Throne.
Reaching King's Landing, Lord Eddard Stark is greeted by Ser Jaime Lannister in the throne room, who recalls listening to Eddard's father and brother scream as hundreds watched them die at the whim of the Mad King. He suggests his assassination of the Mad King was an act of justice, but Eddard dismisses that as a paltry self-justification for breaking his solemn Kingsguard vows.
Rickard is indirectly mentioned together with his son and heir Brandon on two separate occasions to Daenerys Targaryen, the Mad King's last surviving child.
Barristan Selmy vaguely alludes to their deaths when he tells her about how her father murdered sons in front of their fathers and how his actions led to a rebellion that killed every Targaryen, save for Daenerys and her other brother Viserys, who has also since died.
Later, when Tyrion Lannister is advising Daenerys about her strategy of reclaiming the Iron Throne, he points out her lack of support by the Great Houses. As he lists them, he reminds her that House Stark is gone due to the actions of their "two terrible fathers" though he doesn't discuss the specifics - her father executing Rickard and Brandon and his father orchestrating the Red Wedding.
Bran Stark, with the aid of the Three-eyed Raven, has a greensight vision in which he witnesses Lord Rickard advise the young Ned Stark prior to departure from Winterfell to be fostered in the Eyrie. Rickard tells his son not to get into fights, but that if he does he should win them
During his post as Lord of Winterfell, Rickard Stark arranged the mariages of his son Brandon to Catelyn Tully, and his daughter Lyanna to Robert Baratheon. Both engagements foundered when Rhaegar Targaryen stole off with Lyanna, thus initiating a series of events that resulted in the executions of both Rickard and Brandon at the command of King Aerys Targaryen, the Mad King.