One day, Hamish Macdonald sat down in his local cafe and read the morning newspaper. The front page story tells of a series of gruesome murders, and the police had finally caught the killer. Disturbed by the news, Hamish thinks to himself "I bet it was an immigrant. No Scotsman would ever do such a thing." The next day, Hamish returned to the cafe and read the morning newspaper. This time, more details had been announced. The murderer was Angus Balfour, a native of Scotland. Hamish refuses to believe the newspaper, saying "no true Scotsman would ever do such a thing."
Above is a classic example of the appeal to purity fallacy, which, thanks to this story and its variants, is more commonly known as the "no true Scotsman" fallacy. In the example, note how Hamish quickly amends his original statement. He begins with "no Scotsman" would do this, then when it turns out a Scotsman really was involved, he adds one little word: "no true Scotsman" would do this.
The no true Scotsman fallacy, as seen in the above example, is the self-serving redefinition of a term in order to exclude inconvenient examples. To give a more realistic example, consider Apostate Prophet. Apostate prophet is an ex-Muslim who now dedicates his time to exposing and opposing Islam. Because of his status as an ex-Muslim, he is in a particularly useful position to oppose it. This, however, is inconvenient to Muslim Dawahgandists, who frequently attempt to cast doubt on his conversion story. He was never a true Muslim, they claim. This is the no true Scotsman fallacy, because there is no reasonable definition of "Muslim" that would exclude Apostate Prophet during his Muslim years.
It is important to note the phrase "reasonable definition". It is insufficient to simply say "no true X". To give an example, consider the statement "no true circle has 4 equal, right angle corners". A circle, by definition, has no corners. What I just described, in reality, is a square. Circles do not have corners. Therefore, you cannot cite a square as an example of a circle that is inconvenient to a well-rounded case, and it is not the no true Scotsman fallacy to exclude the square on the basis of it not being a true circle.
This is important, as often, Christians are accused of committing the no true Scotsman fallacy due to their exclusion of certain heretical groups or individuals. However, "Christianity" is a solid term which, by its very nature, excludes certain beliefs. It is not the no true Scotsman fallacy to say "no true Christian is an atheist", because Christianity is an inherently Theistic religion. You are not a Christian if you are among the surprisingly high number of atheists who claim to be. Similarly, the Bible does actually address the issues of false conversion, and pseudo-Christianity. It even says it is possible to follow "another Jesus". But the logical law of identity necessitates that those who are following "another" Jesus are not following the true Jesus, and thus, by any reasonable definition, are not true Christians. This is not the no true Scotsman fallacy.
The easiest way to recognise the no true Scotsman fallacy is if the argument can be summed up as "we have a 100% success rate, except for this failure. We won't count that." Whatever is not counted must be reasonably excluded from the category in question. If it is, as with our square that is not a true circle, it is not the no true Scotsman fallacy. If it isn't, such as Angus Balfour, the no true Scotsman fallacy has occurred.