This is a summary, rather than a quote from the book.
Looking into several rooms in a hallway in the House of the Undying, Daenerys sees several scenes, some which appear to be of the past, some of the present and some of the future. The first of this is a woman being savaged by four little rat-like men.
It is commonly accepted that the woman in the first room represents the Seven Kingdoms and the rat-dwarves represent the kings squabbling over the land. That there is only four rather than five of them suggests that at the time of this vision Renly was already dead.
The second is a room filled with savagely slaughtered corpses which appear to have been attending a feast. A dead man with the head of a wolf, wearing an iron crown and holding a leg of lamb like a king might hold a sceptre, sits at a throne, his eyes looking at Dany with mute appeal.
The third room shows her old home in Braavos and Ser Willem Darry.
The fourth shows an old man on a barbed throne, saying to another man below him "Let him be king over charred bones and cooked meat," and "Let him be king of ashes.".
The second room would appear to have been a vision of the future events of the Red Wedding, whereas the third room simply appears to show her a glimpse of her past and the fourth room appears to have shown Aerys, just before his death and the Sack of King's Landing.
The fifth room, finally, shows a man very much alike her brother Viserys, except that he is taller and has eyes of dark indigo rather than lilac. He is speaking to a woman who is nursing a newborn babe, telling her that the child's name should be Aegon and saying that "What better name for a king?". The woman asks him if he will make a song for the child, and he replies that he has a song and that "He is the prince that was promised, and his is the song of ice and fire.". He appears to look at Dany then, as if seeing her, and then he adds that "There must be one more," and "The dragon has three heads.".
The fifth room shows Rhaegar and Elia with the newborn Aegon, as confirmed by Martin. Clearly this vision reveals that Rhaegar saw his children as a fulfillment of prophecy, especially when one realizes that he named the first two after Aegon the Conqueror and his sister Rhaenys; Daenerys later comments on there having been no Visenya, so she seems to have understood Rhaegar's expectation of a second daughter. The prophecies of the prince who was promised seem to tie directly to the prophecies concerning Azor Ahai and the coming of a dark time which Melisandre speaks of. In A Feast for Crows (IV: 520), Maester Aemon reveals that Rhaegar at first believed he was the promised prince due to the burning of Summerhall on the day of his birth and the salt of the tears for those who died there, but that he then became convinced that it was his son Aegon who would fulfill the prophecy.
Most importantly, Rhaegar re-iterates the idea that the dragon has three heads. If true prophecies must be fulfilled -- and not all readers agree that this is the case, citing the apparently-failed prophecy of the dosh khaleen concerning Rhaego -- then it's important to discern who they are. It seems obvious that Daenerys must be one, not least because Aemon points out that "prince" is an error of translation, that among the Valyrians the term was as fluid in terms of gender as the gender of dragons was, and could just as well apply to a princess (IV: 520). But who are the others? Martin has stated that Rhaenys is definitely dead, but also that the three heads need not be Targaryens.
Using these points of data, strong contenders for a place as one of the heads of the dragon are Jon Show (either as the son of Rhaegar and Lyanna, or simply as Ned Stark's bastard), Aegon Targaryen (although see our entry "Is Aegon Dead?"), Tyrion Lannister (either in his proper person or as Aerys's son after a rape or seduction of Joanna Lannister), as well as Bran (due to the suggestion that he might be able to skinchange into a dragon, satisfying the three-eyed crow's promise that he will fly), Arya or even the Hound.