What does that mean, though, in this context?
It means that the rules exist independently of their operators. One can DM well, or not well, and it's still DMing either way.
DM fiat of the kind Aenghus describes is usually fairly obvious, and usually bad for the game in both the long and short term. Do you mean it like "it's your funeral!"? DMs who use fiat like Aenghus said should honestly not expect their game to last long. If it does, that's luck (or player desperation), not judgement, I'd suggest.
That may be. However, in practice I think it's not only possible, but far more likely, for a DM to fail because he doesn't exercise the power of his position.
Even where some rules may be poorly written, gameplay problems will arise only when the DM fails to act.
If the PCs are low level nothings smelling of manure, my default is they don't get a diplomacy check, they'll be turned away at the gates.
If they are obviously rich high level PCs, my default is they get the diplomacy check (or may automatically succeed depending on their approach).
This is of course modified by preestablished history and context, and most importantly by the actions of the players.
For example, this is the sort of ruling that keeps mechanics under control. It may be that "Diplomancy" is too broadly written or the DC's are not appropriate or modifiers are too easy to pump, but that won't matter if the DM uses the skill in a rational manner, rather than just allowing the player to say "I Diplo him" and receive the benefits of the skill regardless of the situation.
The weaker PCs, even if they load up on synergy bonuses and get a ridiculous Diplo mod, will not suddenly take over the game world, because by and large they are not that powerful, and the world recognizes their status. Conversely, high-level PCs may very well be treated as if they are important, because they are. Thus, their opportunities to influence people will be broader. None of this is stated in the text for Diplomacy, but it is common sense.
I would tend to say that a ruling that is arbitrary, unilateral and ignores relevant context is a bad ruling.
And so what we have here, the conclusion that I've alluded to above, is that this is really what people mean when they say "fiat". Not an exercise of power, but an exercise of power accompanied by a negative value judgement about the motivations behind it.
So, even if the player rolls a natural 20 on his attack, and the DM declares that the attack misses, that isn't "fiat" if he has some good reason for it. Conversely, even if he's just applied a circumstance bonus to something, or made a player use Bluff instead of Diplomacy, it may very well be "fiat" if that decision wasn't warranted by the situation. The distinction is not between process and outcome, it's between wisdom and foolishness.
And in that sense, there is "fiat". DMs make bad decisions all the time. The solution is not to say that they shouldn't be the decision-maker, but to communicate and give feedback and push them towards making more informed and more sensible decisions.
This is further confounded, of course, by the assumptions that certain posters make that DMing is essentially in bad faith, and they like to use this "fiat" term to make blanket judgments about DMing decisions they don't like, but which aren't objectively wrong (and indeed in some cases are quite wise).