The only three Targaryens we've actually seen on screen are Viserys, Daenerys and Aemon (who revealed his lineage to Jon Snow towards the end of Season I).
There are some differences in the books. Dany is much younger in the books, for one thing. Her brother is a pompous jerk, for another. But we already knew that. We know that they fled Westeros and came across the Narrow Sea to Pentos, where they sought refuge with Magister Illyrio Mopastis, as wealthy merchant.
We know that their father, the Mad King, Aerys I, was murdered by Jaime Lannister (hence, the nickname 'Kingslayer'). We know that they had an older brother, Rhaegar, who was married to Elia Martell. They had two children. Rhaegar was a melancholy, but studious young man who showed great promise. He was in love with Lyanna Stark. We all know what happens next: the crowing ceremony that took place at a tournament, in which Rhaegar crowned Lyanna Stark the 'Queen of Love and Beauty' instead of his lady wife. Oh, and Lyanna just happened to be engaged to Robert Baratheon at the time. I'm sure that action set more then a few tongues wagging! What happened after is anyone's guess, really. Rhaegar and Lyanna disappeared for quite some time. Tragically, Rhaegar was killed by Robert, after Lyanna had died.
Rhaegar's wife and children were killed by the monster, known as the Mountain that Rides, Ser Gregor Clegane. That filthy and vile man assaulted Elia and killed her, then proceeded to kill the young children. It's no wonder why Targaryen loyalists whisked the two remaining children out of Westeros.
Viserys is a power hungry, condescending man who thinks of nothing but revenge. He wants the Iron Throne back. I'm glad that he didn't get the chance to retake it because his cruel nature and petty character would've translated to worse things, should he have actually sat on the Iron Throne. He would've been a nightmare. I had an inkling the first time I saw Maester Aemon, the first time I caught a glimpse of him at the Wall. Yes, I know he is old and blind, but he had that pale, ethereal quality and I couldn't help but wonder....could this be a long lost Targaryen relative who sought refuge in another part of Westeros? The Wall is a pretty far place to go!
I'm glad that Daenerys was married off to Khal Drogo, because the ancient practice of Targaryens marrying brother to sister is sickening, if you ask me. The idea of incest is so abhorrent!
The Targaryens are known for their pale complexions and silvery white hair, along with blue or lavender coloured eyes. Daenerys sure is a beautiful woman. I'm just sad that she never got the chance to rule with the Dothraki tribe, alongside her husband, who fell ill and died, because of a witch. Her unborn son, Rhaego, died prematurely. It tugged at my heart strings that this poor girl, who had already been through so much, seemed as if she was destined to live a life of sadness and depression.
Daenerys appeared to be a tough cookie, come Season II. If you're anything like me and have read four of the books so far, you'll agree with me that whatever doesn't kill you, only makes you stronger and this adage seems to apply to Daenerys Targaryen. She walked into that funeral pyre at the end of Season I, only to survive and come back out unscathed, with three brand new baby dragons. Season II saw her travelling a great length of the Red Waste, onto Qarth, battling intrigue and deception but hey! She managed to come back out on top!