Fiat Campaign

The most-involved form of wargaming campaign, as well as the earliest; Kriegspiel and Little Wars both used a fiat system. Most wargames in the professional world - i.e., those the military and businesses run - also use some combination of fiat and computer simulation to predict outcomes.

Mechanics

The players and/or referees sit down, discuss the previous game, and project an outcome. That's pretty much it. Fiat campaigns put an extraordinary load on the referees, and are by far the most difficult campaigns to balance well. They're also the most drama-prone, since there's no one to blame but yourself and the ref when you lose. And we all know how much That Guy likes to blame himself.. and That Referee isn't much better. Since there are no real mechanics, a biased system or referee becomes impossible to contain without walking off or forcing them out.

That said? A fiat campaign takes almost no setup. With a reliable, dispassionate GM and mature players, they can be incredibly awesome. But when they go bad, they go legendarily bad. So fiat campaigns have a very bad reputation among gamers, no matter what the individual may call it. The best way to run one is with several judges, none of whom are playing in the campaign itself; this is how the military keeps their own campaigns under control. But no gamer really wants to sit out of the rest of the club's activities for days or weeks of real time.

As hinted above, most of the other campaign systems are a way to pawn off the inherent drama when people disagree about what "should" happen, decrease the massive workload so your GM(s) don't burn out, and keep your slightly-bad apples from completely destroying a group.

The Dirty Secret

Almost any game with a GM has some degree of fiat. The GM will need to look out for mechanics or systems that aren't serving the players, arrange special events to help keep players interested, and generally tweak things to keep players engaged. In a system like a Hybrid Ladder Campaign with randomly-generated enemies, the GM may need to tweak the OPFOR to better balance the mission. Same goes for complex games like Battletech, which have known issues in the unit cost mechanics. Points-less historical wargames like Ambush Alley literally require the GM to do so.