On their way south, Robert and Eddard discuss politics and history, which makes this a very important chapter and kind of hard to summarize.
Once again, just go read it.
But here are the super important things to know:
Ser Jorah Mormont is a spy and he reports to Varys, King Robert's spymaster. He's spying to win a royal pardon so he can come home from his exile; and he's in exile because he tried to sell some criminals into slavery, and Eddard caught him. So Eddard and Jorah don't get along.
Robert wants to assassinate the last two Targaryens, even though they're basically children.
Robert has given more power to the Lannisters by naming Jaime the Warden of the East; this makes Eddard uncomfortable because the Lannisters always seem hungry for power.