Variously called a parable, a principle, and a paradox, Chesterton's fence might best be considered a guiding rule in the field of advanced common sense. It is, in short, the rule that you should not change a thing unless you know why the thing is the way it is.
"In the matter of reforming things, as distinct from deforming them, there is one plain and simple principle; a principle which will probably be called a paradox. There exists in such a case a certain institution or law; let us say, for the sake of simplicity, a fence or gate erected across a road. The more modern type of reformer goes gaily up to it and says, “I don’t see the use of this; let us clear it away.” To which the more intelligent type of reformer will do well to answer: “If you don’t see the use of it, I certainly won’t let you clear it away. Go away and think. Then, when you can come back and tell me that you do see the use of it, I may allow you to destroy it.
This paradox rests on the most elementary common sense. The gate or fence did not grow there. It was not set up by somnambulists who built it in their sleep. It is highly improbable that it was put there by escaped lunatics who were for some reason loose in the street. Some person had some reason for thinking it would be a good thing for somebody. And until we know what the reason was, we really cannot judge whether the reason was reasonable. It is extremely probable that we have overlooked some whole aspect of the question, if something set up by human beings like ourselves seems to be entirely meaningless and mysterious. There are reformers who get over this difficulty by assuming that all their fathers were fools; but if that be so, we can only say that folly appears to be a hereditary disease. But the truth is that nobody has any business to destroy a social institution until he has really seen it as an historical institution. If he knows how it arose, and what purposes it was supposed to serve, he may really be able to say that they were bad purposes, that they have since become bad purposes, or that they are purposes which are no longer served. But if he simply stares at the thing as a senseless monstrosity that has somehow sprung up in his path, it is he and not the traditionalist who is suffering from an illusion."
-- G. K. Chesterton’s 1929 book The Thing, in the chapter entitled "The Drift from Domesticity"
This is a sneaky rule, because it is not usually the case that something goes wrong because of misunderstanding the roles of system components. Change is usually enacted by intelligent people who know what they are doing... which is probably why Chesterton focused on politics, as this is perhaps the area where the term 'expert' is most disparaged.
Even in the realm of politics it is fairly difficult to find clear examples of people getting rid of laws and policies that they don't fully understand, and finding that this was a bad idea. Most legislation is a matter of piling on new rules and regulations, which often have unintended consequences, but are not technically examples of Chesterton's fence. Others are borderline, such as Mao Zedong's elimination of traditional economic and social structures during the Great Leap Forward, which resulted in large numbers of people trying to do skilled labor for which they were not qualified. Of course, what caused the most trouble was Mao's placement of new roles, not the elimination of the old.
Legal reform tends to follow social reform, with the effect that in most modern cases the fence has been well inspected and defended before it is dismantled. Allowing interracial marriage and then gay marriage, allowing black men to vote and then women to vote, and even allowing people to buy and sell gold all had unintended consequences, but weren't really cases where Chesterton's fence applied. People knew what they were doing, and argued about it long and hard before doing it.
However, it is true that the average voter or politician does not tend to focus on the origin of fences. Most debates and media reports focus on tradition and unfounded fears and personal angst. The average citizen is presented with the history and use of fences in such a scattered fashion that it's easy to miss that they are even part of the debate. While somebody somewhere is working hard on understanding today's fences, someone else is working hard on fabricating imaginary reasons for each and every fence -- and the imagineers are the ones who will be most motivated to broadcast their results to the masses.
Once we move out of politics and into science and economics there are important cases that are fence-like, but are perhaps too abstract to be fences. If you kill too many wolves, you are overrun by deer. If you outlaw inflation and the use of foreign currency, hyperinflation runs rampant. If you ban the free market, the market becomes unresponsive to consumers. But no one actually erected the predators, planned inflation, or mandated the free market. These are emergent systems that you should be very careful in modifying, but they can only be considered fences in the most overextended of metaphors.
Unfortunately, Chesterton's fence is most often referenced as if it were a meaningful counterargument. It is important to remember that rules are in place for a reason, but "maybe you missed something" is not an argument, it is simply suggesting that you suspect an argument might exist. This is particularly an issue when dealing with problems that don't have a simple etiology (taxes, marriage laws), or are that emerging properties rather than planned properties (inflation, structural unemployment).