Update June 2011: Mark Addy was awesome as Robert Baratheon. He stood out in the scenes he was in and deserves awards for his performance as Robert.
While reading the books, I imagined Robert as Julian McMahon, the Nip and Tuck guy. He was supposed to have been handsome in his younger days who let himself go when he got older.
Robert Baratheon is the eldest of three sons and was crowned King of the Seven Kingdoms after winning the War of the Usurper, also known as Robert's Rebellion.
Like Eddard Stark, he was fostered at the Eyrie by Lord Jon Arryn. Robert's parents were lost to a shipwreck during this time, making him lord of the Stormlands. He also became betrothed to Ned's sister Lyanna, whom he loved deeply. When Rhaegar Targaryen apparently kidnapped Lyanna, Robert was enraged by the abduction, and by the thought of Rhaegar raping her. Though Robert won his war and killed Rhaegar personally on the Trident, Robert did not get Lyanna back, as she died under unknown circumstances. Robert was thereafter haunted by her memory, and bore an irrational hatred against Targaryens in general and Rhaegar in particular.
In the aftermath of the war, Robert assumed the Iron Throne of the Seven Kingdoms on the pretext of a distant relation to the Targaryens (his grandmother was the daughter of Aegon V). In lieu of Lyanna, he married Cersei Lannister to cement her family's loyalty to his rule. He pardoned many of his former enemies and, in fact, won some over as allies. He named Jon Arryn his Hand of the King, and, not bothering to hide a contempt of politics and court intrigue, left much of the governance of the realm in Arryn's hands.
When Jon Arryn died suddenly, Robert journeyed to Winterfell to meet the last person he truly trusted, Eddard Stark.
Robert was a veritable giant of a man, standing six and a half feet tall, broad chested, muscled, and clean shaven, during his youth. His strength was evident in his wielding of a warhammer lesser men (such as Eddard Stark) could barely lift. However, after years of sedentary ruling and participating and feasts and celebrations, he gained thirteen stone and grew a coarse beard to hide his jowls. The mighty warrior who led Robert's Rebellion and defeated Rhaegar Targaryen at The Trident was gone, replaced by a fat and unhealthy man who closed his eyes to anything he didnt want to see.
He had fathered his first bastard before the age of sixteen, and sired more over the years. He was known for his gregarious, outgoing nature, and was frequently able to turn foes into friends; he was also, despite his philandering ways, deeply in love with Lyanna, having started a civil war solely to get her back. Instead he won a kingdom, a throne made of swords, Cersei Lannister, and a crypt for his beloved, which he still paid respects to long years after her death; ironically, he was more loyal to his dead first love than his actual wife.
Julian McMahon played the part of Robert Baratheon - in my head, anyway, as I was reading the book series A Song of Ice and Fire. He's supposed to be fat and bearded but was good looking while he was young.