An Admin Fiat is when a TV Tropes administrator declares that a rule is in effect as regards certain kinds of page actions. This declared rule is considered site policy, and not up for debate. When an Admin endorses a certain position, the supporters of that position take it to mean they now have the permission to enact any changes in line with the Admin's wording.
This term requires a bit of contextualization, as you may guess, since this sounds terribly authoritarian on its own. First, the site users who can issue a fiat are a very limited- Fast Eddie, Janitor, and Gus are the founding members of the wiki, and the only ones with the authority to declare a fiat. Gus's ability to make a declaration is admittedly theoretical, as I never actually saw him make one and it's likely that the principle of fiats was only developed after he left.
The second piece of relevant information about fiats is that they don't generally happen all that often. The Admins have gotten somewhat wary of making statements that can be interpreted as fiats. This is because of the third point- the fiats weren't a conscious effort on the part of the Admins to institute authority. Rather, they were simply interpreted that way by common tropers.
It needs to be understood- given the situation with Character Named Tropes, the wiki was at an impasse. Arguing about whether using a a character's name in a trope's title to signify meaning had become a constant bone of contention in YKTTW. Then one day, almost casually, an admin posted in a YKTTW "New Character Named Tropes are never justified". And this ended up changing everything. If it's a regular troper making such a statement, well, whatever. All tropers are created equal, and thus we can debate. But the Admin's created the site- they're the ones who presumably know what TV Tropes is really about, and so their judgment is final.
The wariness Admins now feel about making further statements is understandable when you consider the TV Tropes Home Page- they're not kidding when they say they're a buttload more informal than Wikipedia. The fiats were brought about mainly because too much informality ended up being stifling- as the site grew, we needed some sort of basic standards to keep new pages from degenerating into incomprehensible messes of in-jokes. At about the same time the Character Named Tropes fiat was issued, we had some of the earliest problems with duplicate tropes- tropes that got through the YKTTW process without anyone realizing that the trope was already on the site because the name of the first trope was that terrible. An Admin decision on this issue was actually welcomed because it quickly fixed a lot of uncertainty on the Character Named Tropes issues.
As the wiki today has many problems regarding what "official" policy is (mostly because no one wants to defy the informal environment by making serious rules), direct Admin statements even from forum threads that are several months old are treated with all the seriousness of the US Constitution. This can be a little absurd at points, since one admin (Fast Eddie) is still on the site, and he could give specific answers if he so wished. It's just that most of the time he prefers not to, since there's already this impression that he's a dictator which he tries to avoid encouraging if at all possible.
In light of this the name is somewhat awkward- it's impossible to discuss admin fiats without making the admins sound like abstract, all-powerful deities. I don't entirely recall where it came from- it's possible that I was the first one to actually describe them this way and it just caught on. In any case, whatever they're called, they're an important part of the site. There's no meaningful appeal process in TV Tropes, and without fiats a lot of work would be made a great deal more difficult. For what it's worth, the admins don't like that their opinions can be interpreted as absolute law. It's really something that just can't be helped.