Daenerys Targaryen is the youngest survivor of the Targaryen family, the royal family that ruled the Seven Kingdoms until Robert Baratheon and friends kicked them out (by, you know, killing most of them.) Her older and jerkier brother is Viserys; and Viserys and Illyrio Mopatis arrange for Daenerys to marry Khal Drogo, a powerful leader. That works out surprisingly well (though Drogo does kill the abusive Viserys), until Drogo dies. Then Daenerys has to figure out how to make it on her own – just her and her newborn dragons.
Murphy's Law is as follows: Anything that can go wrong, will go wrong. Boy does our princess know how that feels. The central lesson of Dany's life is that things don't always work out the way you expect them to; and usually, they end up worse.
So let's see: Dany was born a princess, but she never got to live like one; instead, she lived as a beggar with her older brother, Viserys. Her brother taught her all about their family history, but he was also physically and verbally abusive. When her appointed husband, Khal Drogo, tries to give her something that she thinks she wants (the Seven Kingdoms), she ends up losing what she already has (Drogo, her baby, and their khalasar). The list goes on.
All this makes Daenerys an underdog hero of this book: she's smart, she's gentle and kind, she's even kind of humble. But wait a minute: terrible things happen because of her. Shouldn't we dislike her? No: in fact, she's our favorite "exiled princess teenager married to a barbarian chieftain who comes to find her place in her adopted culture" character ever.
She reminds us how hard it is to be a hero when your actions can have totally unexpected consequences, or when nice things you do may lead to bad results. Think about the whole Mirri Maz Duur situation: Dany saves the wise old woman from a horrible fate, but Mirri chooses revenge on her people anyway. We can't blame kindness, so we have to blame, um, Murphy's law?
But Daenerys isn't just a lesson; she's also a very engaging character. (So engaging that her chapters were printed as a separate novella and won a Hugo Award to boot.)
Who doesn't love a thirteen-year-old who transforms from meek child into powerful queen? (And yeah, that's our reality check for this character: she's thirteen at the beginning of the book [4 Daenerys 1.4]; and only fourteen when she's pregnant [24 Daenerys 3.91].) Daenerys is smart – definitely smarter than her brother. For instance, when her brother just expects Illyrio Mopatis' help, Dany is perceptive enough to see that Mopatis' tiny smile means something else (4 Daenerys 1.37). She's also gentle and generous, trying to prevent rape (62 Daenerys 7), and even offering her most prized possessions – the petrified dragon eggs – to her brother after he tries to steal them (47 Daenerys 5).
But let's be clear: she's definitely not considering all angles: even when she's trying to protect the Lhazareen women from rape (62 Daenerys 7), she's doing nothing about the fact that Drogo is raiding this peaceful village and selling all the men into slavery for her own sake. And if she gave her idiot brother the eggs, he might buy an army, which would lead to war in the Seven Kingdoms and lots of people would die.
So we cheer when Daenerys succeeds and grows as a character, but we also shudder. Yay, she hatched dragons – but who is she going to (inadvertently) kill with those dragons?
Viserys is Daenerys' cruel brother who dreams of reconquering the Seven Kingdoms. He meets a pretty poetic end, but that doesn't quite make up for the awfulness he was in life. We hate Viserys. Really. He is cruel and stupid and has an inflated sense of entitlement (that means he thinks he can do/have anything he wants).
This guy is kind of like a bad version of Jon Snow (could we call them foils, perhaps?). Both of them are outcasts, although they have both also been given some advantages along the way. That is, Jon Snow has been taught to fight; whereas Viserys has been helped by various well-wishers, such as Illyrio Mopatis. (True, Mopatis expects to be repaid, but still, Viserys is housed in a palace when he's really got nothing.) The main difference is that Jon Snow cares about other people whereas Viserys only cares about himself.
Honestly, weren't you a little happy when Viserys was killed and we didn't have to deal with him anymore? It's a good lesson for all of us: don't be Viserys.
These three servants/slaves are given to Daenerys as gifts from her brother (though really paid for by Illyrio). Each is meant to teach Dany something important: Irri teaches her riding, Jhiqui teaches her the Dothraki language, and Doreah teaches her how to have sex. So, pretty much, they'd all be replaced by YouTube in a modern version of this book. Although they are servants/slaves, they are very loyal to Daenerys, staying with her throughout the book; and they help Daenerys (and us) adjust to her strange new culture.
At first, these Dothraki bodyguards are just bodyguards, not bloodriders like Khal Drogo has: they don't live and die with Dany. But they are loyal to her: think about how they hide the fact that Drogo is weak from the rest of his khalasar (65 Daenerys 8). And at the end of the book, after Quaro has been killed, her three bodyguards accept the role of bloodrider to Daenerys. Since women don't usually have bloodriders, this tells us that something big is happening in this world (73 Daenerys 10).
If Varys is the most devious man on the continent of Westeros, he should really get along with Illyrio Mopatis, the most devious man on the eastern continent. Illyrio Mopatis is one of the Magisters of the city of Pentos, which at the very least means he's an incredibly rich merchant with contacts all over the world (4 Daenerys 1.6). Mopatis may be fat and very rich – so rich that he can buy three petrified dragon eggs – but he still wants more, as can be seen by his scheming with Viserys.
Mopatis doesn't show up a lot in this book (basically, he just gets Daenerys married to Drogo and disappears) but he basically sets that whole plot in motion. And, theoretically, if Viserys/Daenerys took back control of the Seven Kingdoms, Illyrio Mopatis would also get a huge reward. Which really raises the question: how much money does one person need?
Now wait a minute: could Illyrio Mopatis be one of the two conspirators that Arya overhears in 33 Arya 3? One of the men overheard by Arya has a Free Cities' accent and Mopatis is from the Free City of Pentos; both men have forked yellow beards and are extremely fat (33 Arya 3.40); both wear many rings (4 Daenerys 1.26). So, yeah, that sounds like it could be Illyrio Mopatis. This is one of those times where you might wish that George R.R. Martin just gave us an omniscient point of view.
None of these characters show up in the book except in flashbacks or memories. But although they are dead, their effects can still be seen:
(1) The fact that Robert is king is in large part due to how crazy King Aerys was. After all, you need to be pretty crazy to turn more than half the continent against you.
(2) Prince Rhaegar killed Lyanna Stark; and since the woman he loved was dead, Robert Baratheon was free to marry Cersei Lannister. (Although, maybe Robert didn't need to tell Cersei that she was competing with a dead woman who was so much better.)
(3) Jaime Lannister was in the Kingsguard – the group sworn to protect the king – so it's pretty significant that he was the one who ended up killing the Mad King. That's one reason why some people don't trust Jaime (31 Eddard 7.143).
(4) The fact that all of these Targaryens died means that Viserys and Daenerys are on their own; and since most of them died violently, we might be a little nervous about what's going to happen to Daenerys, the last Targaryen (and the only one we like).