As most readers are no doubt aware, American forces were able to track down and kill Hussein's two sons, Uday and Qusay. I doubt they'll be missed too much by either the Iraqis or the Americans, and it may well provide a boost for the President's flagging credibility, as he can show that he actually has accomplished something that both the administration and the American people believed was a key goal of the war.
The real story, though, is what happens in the days ahead. There are multiple interpretations of the nature of the guerilla war being waged against the American occupying forces, but one of the most popular right now is that Saddam and his former friends and allies are behind the attacks.
If that is true, then it would stand to reason that the attacks will change dramatically in intensity and numerousness in the days ahead- either they'll dwindle due to the perception that Saddam's regime is well and truly over and the Americans are there to stay (and due to messed up C&C), or increase due to anger among Saddam sympathizers that the Americans had killed the two most important Ba'athists after Saddam himself, including his protege Qusay.
If there is no real change, however, then this implies that the resistance is not solely Saddamist enterprise, but is much more populist in nature. Whether the Hussein brothers are alive or dead is unimportant except in a symbolic sense, and the possibility exists that while the Iraqis don't want the Americans to remain, they aren't particularly sorry that the brothers are dead. Sort of a "pox on both their houses" reaction.
While I doubt that this particular analysis will be a part of the public debate on this, I think it will be indirectly understood depending on what happens. If the American public sees that attacks continue despite the Hussein brothers deaths, then they'll start wondering what exactly will calm things down. If they believe the answer is "nothing", then support for the war, and for Bush, will likely continue to drop. If the attacks do relent, then it's likely that the public will get back onside with the Bush administration.
In the meantime, though, I'd expect a spike in approval ratings.