They ruled over Westeros for several hundred years, first because they were the only family that could command dragons, and later because of the weight of tradition. They were, of course, the Targaryens, once the most feared family in the Seven Kingdoms.
By the time of the events of the series Game of Thrones, the numbers of the direct line of this royal family are severely reduced. While the series does mostly adhere to how the characters are described in the original books, there are some key differences with certain characters.
Daenerys
Unquestionably, Dany is the most important surviving member of the dynasty, living in exile on Essos while scheming to regain the throne of Westeros. In the novel she has the appearance common to almost all members of her family: white-blonde hair, violet eyes, and a thin frame. Depicted by Emilia Clarke in the television adaptation, her appearance in the series is largely in keeping with the way that she is described in the novels, however, her eye color didn't make it onto the small screen. Via Mewlish.
Maester Aemon
Though at first, it isn’t clear that he is a member of the Targaryen family, at one point the aged maester at the Wall reveals the truth to Jon Snow. Given that, by the time of the series, he is a very old man indeed, the novels describe him as blind, with a thin neck. The television adaptation holds very true to this description, and the man who portrayed the aged maester was himself quite old during filming (in his 80s and 90s). From Pojypojy.
Viserys
Viserys is Dany’s cruel and petty elder brother. During their time on Essos, he ruthlessly abuses her and sells her in marriage to Drogo in order to gain the alliance of the Dothraki. Like his sister, he is described as being Valyrian in appearance, though here the fine bone structure has been turned into a gaunt and fevered look.
The television adaptation shows him in a similar fashion, though he doesn’t appear quite as gaunt or desiccated as his novel counterpart. Art by Adamski.
Aerys
Known as the Mad King, Aerys has the dubious distinction of being such a terrible and inept ruler that he plunged his kingdom into the civil war that would eventually see him dead at his own guard’s hands. As his madness took hold, it began to affect his physical appearance, and he turned from a handsome prince into a king with unwashed hair, untrimmed nails, and a horde of scabs on his legs. The television show doesn’t show him often, and when it does he looks significantly more well-kempt than the novels describe. From KelpyKrad.
Rhaegar
One of the greatest tragedies in Westeros was that Rhaegar, the noble Targaryen prince, wasn’t able to take the throne from his father. With his signature Valyrian beauty, his deep sense of honor and justice, and his pleasant personality, Rhaegar would have been the best Targaryen king in several generations. He only appears in flashbacks in the television series, though what the viewer sees of him is in keeping with the novel, for he is shown as being very beautiful. Via aegia.
Bloodraven
Bloodraven is one of the many characters from the novels that wasn’t translated to screen. He is a bastard offshoot of the Targaryen family tree, and so his looks are not described as being strictly Valyrian. Instead, he is an albino, with white hair, red eyes, and a distinct birthmark on his cheek that gave him his nickname (his real name was Brynden Rivers). In the novels, he is the three-eyed crow, though this connection isn’t made explicit in the television series. Art from aegia.
Jon Snow
Long thought to be the bastard son of Eddard Stark, late in the series it is revealed that Jon Snow is in fact the child of Lyanna Stark and Rhaegar Targaryen. Despite his Valyrian ancestry, he is described as having dark hair and fair skin, with eyes that are a very dark grey. In the series, this is translated very competently, and Kit Harrington looks a great deal like the version of Snow that is described in the novels. Via Majoh.
Aegon the Conqueror
Arguably the most important member of the Targaryens was the man who started it all, Aegon the Conqueror, who swept most of Westeros before him on his magnificent and terrifying dragon.
Though he never appears in the television series, he is still a significant character in its mythos. In the various supplementary material that Martin has published, he is described as being of typical Valyrian appearance, with purple eyes and silver-gold hair. Art from chillyravenart.
Visenya
Aegon did not come to Westeros alone, for he also brought with him his two sisters, one of whom was Visenya, one of the most warlike women to have born the Targaryen name. Like her brother and sister, she had typical Valyrian features, with long gold hair (with some silver) that she usually wore loose. She is described as having a harder, sterner appearance than her sister, more like her brother Aegon. There were rumors that, in addition to being a very sensual and passionate woman she was also a sorceress. By ProKriK.
Rhaenys
Rhaenys was the other sister of Aegon that accompanied him on his conquest of Westeros. Like her sister, she rode a dragon and, like her sister, she wore her long hair loose. She was unlike either of her siblings, however, in some key respects. She was more light-hearted than they were, more fond of music and song and dance. Most importantly, however, she truly loved flying, so it is perhaps fitting that she perished riding her dragon while warring on Dorne. From friedrich-todd.