Theon Greyjoy is the only living son of Balon Greyjoy, Lord of the Iron Islands. We say only living son because back in the day Balon tried to rebel against Robert Baratheon's rule and crown himself the King of the Iron Islands. Let's just say that went about as well as Rhode Island rising in armed conflict against the rest of the United States.
During the crushing defeat, all of Balon's children were killed except for Theon and Balon's daughter, Asha. Theon was sent to live with Eddard Stark as his ward (read: don't rebel again, Balon, or else), and the boy called Winterfell his prison-home for ten years.
We didn't really get to know Theon Grayjoy back in A Game of Thrones, but there is no time like the present to catch up.
You remember the seagulls from Finding Nemo, the ones that were always squawking mine? That's basically what we hear when considering Theon Greyjoy's character.
Catelyn Stark remembers Theon for his "sly smile" and his "way of looking as though he knew some secret jest that only he [is] privy to" (8.Catelyn.28). That secret is that he believes the world is his for the taking, and the sooner everyone realizes this fact, the better life will be—most of all for Theon.
We see this mindset of Theon's in the way he treats the captain's daughter on his journey to Pyke. Sleeping with her, Theon takes what he wants from the girl and gives her nothing in return, dumping her flat after toying with the girl's hopes. He doesn't even bother to learn her name, thinking of her simply as a "stupid girl" (12.Theon.21). Nice.
He brings the same mentality to most everything else. While sailing to Pyke, Theon looks at the red comet in the sky and believes it is his comet (12.Theon.7). Maybe it's just us, but it takes a lot of ego to claim a comet as your own.
While at Pyke, Theon presents his plan to help Robb fight against the Lannisters, all the while believing the victory will surely be his, and that for his reward, he will receive Casterly Rock, rising high in the world (12.Theon.171). When his sister pretends to be the shipbuilder's wife, Theon tries to take her for his own anyway, ending up rather embarrassed for his flirty efforts.
We'll admit that one of Theon's better qualities is his ambition, a desire to prove himself to his father and others (25.Theon.128). But this ambition quickly turns to greed within him time and time again. Whether it is taking a lover, or helping the person he considers a younger brother, all of Theon's thoughts and plans eventually come to rest on one thought: How does this help Theon get what Theon wants?
Having lived ten years away from Pyke, Theon returns home to a cold reunion and thinks: "It is as if I were a stranger here. […] Nothing has changed, and yet everything has changed (12.Theon.133). And in a way, he totally is.
Balon Greyjoy does not trust his son and does not know him enough to love him, plus he believes the Starks have converted him to their green land ways. We see this when Balon balks that his son wears clothes and jewelry he paid for with the "gold price" and not the "iron price" (12.Theon.156). To translate, this means he paid money for the items rather than taking them from someone he'd killed.
Theon's outsider status becomes pretty clear to him as well. While Asha is pretending to be Esgard, even prideful Theon realizes he does not know the ironborn nor their ways as well as this girl, and Theon thinks that he must prove himself before his father dies (25.Theon.128).
But why doesn't Theon just swing back over the Starks? He could just say, "Hey, things didn't work out with my old man. Mind if I crash here for, like, forever?" Well, that's because Theon didn't really feel at home with the Starks either.
Sure, he sees Robb as a younger brother, but he remembers:
Lord Eddard had tried to play the father from time to time, but to Theon he had always remained the man who'd brought blood and fire to Pyke and taken him from his home. As a boy, he had lived in fear of Stark's stern face and great dark sword. (12.Theon.100)
Whether they treated him as an equal or not, Theon was the Starks' hostage, pure and simple. Lord Eddard was his jailer, and he later recalls the relationship as being like a noose tied around his neck (67.Theon.60). Not exactly imagery one associates with happy homes and loving families, is it?
With two father figures in his life, Theon has learned two completely different sets of values, and these worldviews compete with each other within him.
After raiding a fishing village per Balon's orders, Theon thinks, "Paying the iron price. My lord father would approve." He briefly considers searching the two men he's killed for plunder worth taking, but "the notion [leaves] a bitter taste in his mouth," and he "imagine[s] what Eddard Stark would have said" (38.Theon.18). We can clearly see the mental battle going on in Theon's soul in this scene, as though he has a devil and an angel father perched on each shoulder.
We see this further when Theon takes Winterfell. Rather than raiding and plundering the castle like an ironborn should, Theon stays behind and tries to govern the place like the Starks would have. Asha even swings by to tell him he's been an idiot and that the ironborn get their power from the sea, not landlocked castles (57.Theon.44).
Frustrated and having lost any sense of home in the world, Theon doubles down on his mistake to take Winterfell. When Bran and Rickon run away, he kills the miller's boys and passes them off as the young Starks, deciding that it is "better to be seen as cruel than foolish" (51.Theon.62). Still unable to satisfy his ambition, he has a crown made and dubs himself the Prince of Winterfell.
His reign is cut short when Ramsay Snow besieges the castle and proves himself a better ironborn than Theon, pillaging and burning the castle. Theon's fate at Ramsay's hand remains a mystery.