He saw his mother sitting alone in a cabin, looking at a bloodstained knife on a table in front of her, as the rowers pulled at their oars and Ser Rodrik leaned across a rail, shaking and heaving. A storm was gathering ahead of them, a vast dark roaring lashed by lightning, but somehow they could not see it.
This section of Bran's dream-vision clearly refers to the danger gathering in King's Landing.
He saw his father pleading with the king, his face etched with grief. He saw Sansa crying herself to sleep at night, and he saw Arya watching in silence and holding her secrets hard in her heart. There were shadows all around them. One shadow was as dark as ash, with the terrible face of a hound. Another was armored like the sun, golden and beautiful. Over them both loomed a giant in armor made of stone, but when he opened his visor, there was nothing inside but darkness and thick black blood.
The most important thing about the passage above is that it implies that Bran is seeing all of these things concurrently. One might imagine it as if he has three "windows" in his view, one showing his father and Robert, one showing Sansa, and one showing Arya. Around these "windows" are the shadows. The ash-dark shadow with the face of a hound is obviously the Hound, Sandor Clegane. The figure in golden armor must then be the Kingslayer, Jaime Lannister. The giant in armor made of stone has been discussed a little more, however. There are three main possibilities: Robert Baratheon, Tywin Lannister, or Gregor Clegane, the Mountain that Rides. In the case of Robert, the argument runs that the other figures present in the scene are figures whose actions (or future actions) directly or indirectly affect both of Bran's sisters. Therefore, Robert being present makes sense, since he is the one who is the crux of many events.
However, while the "giant in armor" who stands behind and above the other two men suits the tall and powerful king, the "armor made of stone" filled with "thick black blood" doesn't seem to suit Robert very well at all. While he led a rebellion and turned a blind eye to certain unpleasant events, he was not himself a bad man. On the other hand, this does fit Ser Gregor Clegane very well, from the stone armor to the great height to the darkness and the blood that's thick and black (the result of the poison Oberyn Martell used in their duel). The counter to this is that, so far, Ser Gregor has not done anything to impact both of the girls directly. In response, we think that there's time yet for Gregor Clegane to affect Sansa Stark's life in some notable fashion.
Perhaps the best argument for it being Gregor (or, at least, not Robert) is the fact that Robert appears to already be in the vision, since Ned is pleading to him. There's no good reason for Robert to be a "shadow" looming over himself. This would rule out Robert as the giant in stone armor. The other possibility is Lord Tywin, but we personally don't favor this argument. The stone armor might represent the Rock, but the "thick black blood" really seems too strong even for Tywin Lannister, whose actions -- while often wicked -- were committed with a ruthless precision, and were only a means to an end. Of the three candidates, only Gregor Clegane is a brutal butcher who is good for nothing else but killing.
A Dance with Dragons appears to add significant weight to Gregor Clegane as the giant. Because of his apparent death and resurrection, or perhaps near-death and necromantic healing, "Ser Robert Strong" is clearly what remains of Ser Gregor. In his role of Queen Cersei's likely champion, and his place in Tommen's Kingsguard, strongly suggests that he will in fact have a more direct connection to Ser Jaime than he has previously had.
A last possibility that has been put forward by several fans is that the stone giant is a reference to Littlefinger, due to the fact that the Titan of Braavos was the emblem of Littlefinger's foreign ancestors. Littlefinger is in some sense a "giant", thanks to his significant involvement in the events of the series, events which have affected both Sansa and Arya. However, the reference to the "visor" strikes us as something that would rule it out, due to the fact that the Titan of Braavos has a face. Of course, this could just be taken as Bran's vision putting a reference to the Titan in a guise that he would understand (a giant knight in stone armor), but why not an actual stone giant instead? Bran understands this figure just as well as he would a knight, and it would be somewhat more representative of the Titan of Braavos. An objection to this is that it'd be too obvious -- but then, we never hear about the Titan until the third book, so in retrospect the prophecy would have been quite obscure up to that point (at the least). Finally, if the "thick black blood" is too strong for Tywin (as we feel it is), the same could be said for Littlefinger. Only Gregor Clegane seems monstrous enough to have "nothing but darkness and thick black blood" in him.
He lifted his eyes and saw clear across the narrow sea, to the Free Cities and the green Dothraki sea and beyond, to Vaes Dothrak under its mountain, to the fabled lands of the Jade Sea, to Asshai by the Shadow, where dragons stirred beneath the sunrise.
Asshai, one of the fabled exotic kingdoms of the east, is a place of mystery and sorcery. It lies besides the Shadow, an even more mysterious and forbidding land. Bran's vision may well indicate that there are still dragons there.
Finally he looked north. He saw the Wall shining like blue crystal, and his bastard brother Jon sleeping alone in a cold bed, his skin growing pale and hard as the memory of all warmth fled from him ... North and north and north he looked, to the curtain of light at the end of the world, and then beyond that curtain. He looked deep into the heart of winter, and then he cried out, afraid, and the heat of his tears burned his cheeks.
The vision of Jon with his skin growing pale and hard could possibly be somewhat worrying. At first this was thought to be largely a metaphor for the hardships and the isolation from the rest of his family he'll undergo. However, the events of A Dance with Dragons now open the possibility that Jon Snow's body will be preserved in one of the cells beneath the Wall, growing "pale and hard" after life's fled. This may help to bring about a second vision, Dany's in the House of the Undying, in which she is shown a blue rose growing in a chink in the Wall. Given the author, we can't feel fully confident in saying one way or another, but personally our opinion is that the latter is most correct. As to the rest of the vision, our personal theory is that the curtain of light is a magical barrier made long ago to enclose the Others in "the heart of winter". It's also possible that the reference is simply to the aurora borealis.
One suggestion that doesn't work, however, is that the light is a barrier erected by the Others to keep "dreamers" from spying on them. The Others are the antithesis of light, and never appear anywhere near it. Moreover, Bran feels no resistance or difficult in passing the barrier, and most importantly of all there are defenses made against him immediately beyond the curtain of light which seem to suit the cold Others much more properly.
Now you know, the crow whispered as it sat on his shoulder, now you know why you must live.
"Why?" Bran said, not understanding, falling, falling.
Because winter is coming.
This one simply indicates Bran's importance in stopping the coming Winter and the Others.
Bran looked down. There was nothing below him now but snow and cold and death, a frozen wasteland where jagged blue-white spires of ice waited to embrace him. They flew up at him like spears. He saw the bones of a thousand other dreamers impaled upon their points. He was desperately afraid.
See above -- these jagged spires must be the defense the Others have against "dreamers" who spy upon them.