Military Operations on Urban Terrain.
I'm not sure if that's still the name in current doctrine, but that's what fighting in a city was called when I was in the Army (1983 - 1993).
It seems that our military, nominally assisted by Iraqi security forces, is prepared to move into Fallujah within hours or days. There were several bombing sorties by American aircraft overnight and US Marines have stated that the "battle space" is being prepared. In other words, it won't be long.
Whether or not the name has changed, I can tell you what hasn't changed; fighting in a city is dangerous, bloody work. The fighting takes more soldiers that are required to fight on a similarly sized piece of open terrain and many more soldiers than it takes to hold open terrain after the battle.
The defenders always have the advantage, regardless of the disparity in technology. They know the terrain, they know the buildings, they've had time to knock out walls between buildings to provide movement and escape routes. They have the advantage of height, observing and fighting from the tops of buildings. They know and likely have the cooperation of the local population. In more open terrain, an attacker, doctrinally, requires a 3:1 advantage. In a city, especially an older, non-rectilinear one like Fallujah, the required advantage can climb to 5 or 10:1.
Although our soldiers will go into this fight with a huge technological advantage, unless the putative Iraqi government and BushCo. are willing to literally level the city, there is no doubt that this fight will devolve into house-to-house fighting. To do this, each building has to be isolated from the ones around it, and then soldiers fight their way into the building - usually on a floor other than the bottom - and then clear each floor, up and down. Once it's been cleared, a detail has to stay behind to make sure it's not reoccupied.
You can see how the numbers of troops required to do this can climb very quickly depending on how tenacious the defenders are. Throw in their limitless supply of high-grade explosives, looted from al QaQaa, plenty of RPGs and lots of fighters willing to blow themselves up or fight to the death, along with American 250 and 500 lb laser-guided bombs, Hellfire missiles and 2.75 inch aerial rockets and you've got a scene that would have made Dante blanche.
It's no wonder Bush wanted this to wait until after the election.