The theory of fallacies is merely partially systematized folklore; as one would expect from folklore, it is a weird brew of logical tidbits, practical advice, ethical admonition, historical detritus of exploded or doubtful theories, things people thought clever or neat at some point, and misunderstandings. I've given a number of examples over the years. A very obvious example is that of faulty analogy; the 'fallacy of faulty analogy' was put on fallacy lists by utilitarians, to whom it meant 'you are using an analogy that would suggest that utilitarianism is wrong', given a somewhat more rigorous, and less obviously tendentious, account by Mill on the basis of a theory of analogy almost nobody has ever accepted, and from there was copied from text to text, most often uncritically and sometimes with mutations when you get to an author of a critical thinking text who actually engages in critical thinking and starts wondering what explains this being on a list of fallacies. The 'fallacy of special pleading' is another good example of this mess. If you look at serious work in informal logic over the past forty or fifty years, one gets a farcical comedy of baffled intelligent people trying to make sense of its status as a fallacy. Giving an account of it that fits the examples to which it is typically applied seems to make inexplicable why it is treated as a fallacy; giving an account of it that makes it definitely a fallacy suddenly leaves it orphaned, since almost nothing to which it has ever applied then turns out to be an example of it.
As one might expect in these circumstances, there are quite a few different things called 'special pleading', and which one a person is using typically depends on whatever third-rate informal logic manual they got the concept from, or even on a person's vague impression of how the term is used in ordinary conversation.
A very common way of trying to make sense of it is by treating it as a form of double standard: an exception to a general rule is given and not justified. At this level of generality, it's useless; we need more precision about this 'general rule'. For instance, is it a generally accepted rule or a rule that the person in question is accepted. If the latter, that could possibly be a case of someone reasoning fallaciously, but then it just seems to be ordinary contradiction put inexplicably in fancy dress, since the actual fallacy is just self-contradiction. If the former, that would suggest reasons to have a particular category called 'special pleading', but also makes it not a fallacy -- it cannot be an error of reasoning to insist that there is an exception to a rule that only people other than oneself take to be general. The fact that justification is required also seems to suggest that it has to be the person's own rule; but if that's the case, and you are not just contradicting itself, it seems like the only other possibility is that the other party is mistaken in thinking that the rule is in fact a relevantly general rule -- perhaps it is only a rule for a specific domain and was never formulated for the domain of the exception, or perhaps it is a crude rule of thumb, or the like.
Older books often classify it as a particular form of fallacy of accent, which notoriously makes no sense whatsoever and has baffled logicians ever since -- fallacy of accent is an ambiguity arising through prosody (for instance, change of stress), but the older authors seem to be treating it as a confusion of aspects. As near as we can tell today, it seems that the idea was that they were taking it to cover any kind of difference in emphasis. Out of this line of thought comes the account on which special pleading is one-sided argument. Thus if you only give the reasons for something and not the reasons against it, you are special pleading. This actually has an advantage of making sense of the label itself, but as has been pointed out for years and years now, this is not a fallacy, it's just ordinary argument.
As usual, the mess becomes even more obvious when one looks at examples. Here is Nizkor's best example:
The person committing Special Pleading is claiming that he is exempt from certain principles or standards yet he provides no good reason for his exemption. That this sort of reasoning is fallacious is shown by the following extreme example:
Barbara accepts that all murderers should be punished for their crimes.
Although she murdered Bill, Barbara claims she is an exception because she really would not like going to prison.
Therefore, the standard of punishing murderers should not be applied to her.
This is obviously a blatant case of special pleading. Since no one likes going to prison, this cannot justify the claim that Barbara alone should be exempt from punishment.
This is a very odd example. Nizkor takes the fallacy to be one "in which a person applies standards, principles, rules, etc. to others while taking herself (or those she has a special interest in) to be exempt, without providing adequate justification for the exemption". 'Punishment' and 'going to prison' are not synonymous, so the 'exception' would not, strictly speaking, be an exception to this rule in particular. If you go with one or the other, though, Barbara has merely contradicted herself. If the rule is actually general, no exception could have justification; if we are considering whether there are exceptions, then it is an open question whether the rule is actually general, and therefore one should not simply be assuming that it is. Barbara explicitly gives a justification for her conclusion, so the final comment puts all the emphasis on the adequacy of her justification. But why is her justification inadequate? Only because it is irrelevant. But this means that to determine that this is special pleading, we had first to identify a different fallacy that is usually given a different name, and then we called that fallacy 'special pleading' for reasons that are unclear. This latter kind of thing would be possible if, for instance, 'special pleading' were a label for an ethical fallacy, since another fallacy could also be an ethical fallacy in a particular context, but that is not how the fallacy was characterized, and it is not how the example is set up.
Nizkor goes on to claim that the fallacy of special pleading is a violation of the principle of relevant difference -- things can only be treated as differently if they have a relevant difference -- (and this seems to be what FallacyFiles has in mind, as well, although since their primary example is not even a fallacy, it's less clear) but Barbara is identifying what she takes to be a relevant difference. So the problem must be that it's not actually relevant, which means that either she's really committing an ignoratio elenchi (which is a very plausible diagnosis here), or is just wrong -- and it's not a fallacy to be wrong. So, again, it seems that we determine that something is special pleading by first determining that it is, in fact, another fallacy, and then we call it 'special pleading'. A different example just confirms this:
Bill and Jill are married. Both Bill and Jill have put in a full day at the office. Their dog, Rover, has knocked over all the plants in one room and has strewn the dirt all over the carpet. When they return, Bill tells Jill that it is her job to clean up after the dog. When she protests, he says that he has put in a full day at the office and is too tired to clean up after the dog.
This is a very plausible violation of the principle of relevant difference. But while Bill's reason is relevant to why he should not clean up after the dog, it is not relevant to why Jill should do it, because Jill has the same reason. So again, we seem to have ignoratio elenchi.
Perhaps one could say that 'special pleading' is in fact an ignoratio elenchi applied to oneself? But it's not clear why applying to oneself would be an important classificatory factor here. It might be if 'special pleading' is, again, an ethical criticism. And this would explain some things. In practice people tend to use 'special pleading' like they use 'straw man' -- it looks like a neutral term but over and over again one finds that people use it to insinuate bad motives. You are treating yourself as special by not holding yourself to standards to which you hold others! It is always presented as a formal and objective criticism of the argument, but it gets its bite from being treated as if it were an ethical criticism of motivations. Perhaps, then, it's strictly speaking an ethical criticism presupposing a formal inconsistency; but it's difficult to find cases in which people actually use it this way -- the ethical criticism would require information beyond the argument, whereas people treat special pleading as diagnosable from the argument itself.
The IEP gets a much cleaner account at the cost of just making 'special pleading' another name for logical inconsistency, or perhaps for logical inconsistency on matters that interest the arguer, which seems pointless and useless.
If we attempt to think through fallacies critically rather than in the uncritical way most people do, a question that immediately comes to mind is, "How does one determine that something is an example of this particular fallacy and not something that merely looks like it?" If the answers you get amount to, "Prove that another fallacy is committed, or prove that it's wrong," this should be an immediate red flag: the former seems to reduce your fallacy to another fallacy, and the latter makes it not a fallacy, since merely being wrong is not a fallacy. This is exactly the situation with 'special pleading'; the answers are almost always, "Because the person contradicted themselves" or "Because their reason isn't relevant" or "Because they are wrong". If they are contradicting themselves, why not say that? If they are wrong, why not say that? On the other hand, if the complaint is that they are arguing dishonestly, why are you phrasing a complaint about their motives as if it were a complaint about their argument? It's not that you couldn't have good answers to these questions in particular cases; but they don't seem to apply generally. Likewise, maybe a particular instance of 'special pleading' is used meaningfully -- but it looks much like you can't assume that everyone will understand your use in the same way, and it will always be unclear how much of your use is just rhetorical flourish. The whole practice of classifying reasoning as 'special pleading' continues to be problematic.