First, he refuses to follow the King's command to kill Daenerys - the King's command occasioned by the news that Daenerys is pregnant, which cements the alliance of the Targaryen and the Dothraki, and sets up a much formidable threat to the throne. Ned understands the threat, but is too decent to kill a woman with a child - or, for that matter, likely anyone who doesn't literally have a sword headed towards the King's neck. Ned resigns his position as Hand. Will his relationship with King Robert be old and deep enough to survive this?
Next for Ned is indeed a sword-in-hand confrontation with Jaime Lannister, who is furious about Tyrion's being taken hostage by Catelyn. Lannister's men with little warning kill all of Ned's men by hurled spear, except Ned's Captain of the Guard, who's dispatched by Jaime via sword through the eye. Jaime and Ned have a brief, intense duel, ended when one of Jaime's men spears Ned in the leg.
Why did he do that? Likely because he estimated that Ned would prevail in the duel. Jaime - unhappy with this conclusion to the sword play - strikes his man. Ned has already aptly noted that Jaimie can't kill Ned, which would forfeit Tyrion's life in Catelyn's hands.
But this whole life-and-death incident - the best in the episode - raises an interesting question: if Jaime was responsible for the second attempt on Bran's life - we know he was responsible for the first, we saw that with our own eyes - why would he be so outraged at Tyrion's being taken prisoner? That is, Jaime would want Tyrion freed in any case, but there was something in his outrage that suggested (at least, to me) that neither he nor Tyrion was responsible. Further, come to think of it, why would Jaime risk arming Bran's would-be assassin with Tyrion's sword in the first place? But if not Jaime, who then was behind the second attack on Bran? Cersei would similarly not be likely to put Tyrion's sword into the mix.
On other hand, it's still early times in Game of Thrones, and we don't know any of the characters - least of all the Lannisters - all that well ...
See also A Game of Thrones: My 1996 Review of the First Novel ... Game of Thrones Begins Greatly on HBO ... Game of Thrones 1.2: Prince, Wolf, Bastard, Dwarf ... Games of Thrones 1.3: Genuine Demons ... Game of Thrones 1.4: Broken Things
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