What would happen? The same thing that happens in other countries when their central governments fail. Check the nearest history book for details. :)
The only semi-realistic things that I can think of off the top of my head that might cause our federal government to suddenly fail are total economic collapse, all-out nuclear war, or pandemic disease.
In either case, then in the short term, the military (and/or the DHS, here in the US) would very likely assume control, and we’d be a de-facto dictatorship. Depending on the exact circumstances, we’d either then continue as a dictatorship, or a democratic government of some sort would eventually be re-established – probably with a new (or at least modified) constitution, and with significantly less real power, no matter what powers the founding documents ostensibly gave it, with the military retaining significant actual decision-making power.
In the process, there’d be riots, pockets of the country controlled by local forces, economic breakdown, hyperinflation (if that’s not what caused the failure in the first place), food shortages, energy shortages, and general mayhem. Depending on how bad the failure was, there might be civil war – for instance, if the west coast decided to try to set up its own nation, or Texas decided to secede, or if there was large-scale disagreement about how the country should be put back together; fortunately, although we complain about poor race relations here in the US, our problems in that regard are in reality minimal compared to the centuries-old tribal disputes and rivalries that exist in some other parts of the world, so I think full-scale civil war would be less likely here.
But maybe you didn’t mean sudden failure, but rather what would happen if our democracy failed slowly, over the course of decades or even generations?
In that case, you’d also see things here that we’ve seen in other countries where that’s happened: the civil government, though technically still in charge of the country, would slowly lose real power and prestige until some other nexus of power would, pragmatically speaking, actually be in control. In that scenario I think you’d see the federal government become gradually more controlled by powerful and well-funded institutions like the military or possibly very large corporate interests; the government would never really “fail” or “fall”, per se, it would just cede control slowly over time. The end-game in that process tends to be either dictatorship or fascism; in either case, the elections would either get shut down, or become so thoroughly controlled/corrupted that they had little real effect, and the press would no longer free, but rather controlled by the ruling interests, whoever they were.
I suppose an argument could be made that a process much like that is already underway in the US..