Argument from authority or appeal to authority is a logical fallacy, where it is argued that a statement is correct because the statement is made by a person or source that is commonly regarded as authoritative. The most general structure of this argument is:
Source A says that p.
Source A is authoritative.
Therefore, p is true.
This is a fallacy because the truth or falsity of the claim is not necessarily related to the personal qualities of the claimant, and because the premisses can be true, and the conclusion false (an authoritative claim can turn out to be false). It is also known as argumentum ad verecundiam (Latin: argument to respect) or ipse dixit (Latin: he himself said it).
There are some things wrong here. For one thing, argument from authority is not a fallacy but a rhetorical strategy; it can be entirely nonfallacious if the source really is authoritative. And not every argument that is not logically valid is a fallacy, as everyone learns in a good philosophy class. But it's interesting to think about 'argumentum ad verecundiam' (which in context actually means an appeal to the sort of shame associated with modesty), and why appeal to modesty (1) became discussed in this context at all; and (2) became regarded as a fallacy. The two points need to be separated because they are not the same.
Locke introduces the phrase into discussion of arguments in the Essay Concerning Human Understanding, IV.xvii.19. He is talking about "four sorts of arguments, that men, in their reasonings with others, do ordinarily make use of to prevail on their assent; or at least so to awe them as to silence their opposition." The first of these is described in these terms:
The first is, to allege the opinions of men, whose parts, learning, eminency, power, or some other cause has gained a name, and settled their reputation in the common esteem with some kind of authority. When men are established in any kind of dignity, it is thought a breach of modesty for others to derogate any way from it, and question the authority of men who are in possession of it. This is apt to be censured, as carrying with it too much pride, when a man does not readily yield to the determination of approved authors, which is wont to be received with respect and submission by others: and it is looked upon as insolence, for a man to set up and adhere to his own opinion against the current stream of antiquity; or to put it in the balance against that of some learned doctor, or otherwise approved writer. Whoever backs his tenets with such authorities, thinks he ought thereby to carry the cause, and is ready to style it impudence in any one who shall stand out against them. This I think may be called argumentum ad verecundiam.
Locke is not talking about fallacies here. He does deny that an argument ad verecundiam "brings true instruction" or "advances us on the way to knowledge". That is only possible with ad judicium, which appeals to probable and demonstrative reasons. But he does allow that the other three sorts of argument, including ad verecundiam might dispose you to knowledge. These four sorts of arguments are really rhetorical strategies: appeal to the modesty of your interlocutor, appeal to the fact that your interlocutor doesn't seem to be able to find a better argument, appeal to claims that your interlocutor has already conceded, and appeal to "proofs and arguments" about "the nature of things themselves".
This idea for classification seems to have caught on, and throughout the eighteenth century we see authors of logic textbooks expanding Locke's list. Isaac Watts, who wrote one of the important logic textbooks of the eighteenth century (he is the same Isaac Watts who wrote so many classic hymns), has a whole list, including ad verecundiam, that expands considerably on Locke's four. Watts considers ad verecundiam and its kindred labels to be topics, loci communes: they are a way of classifying arguments (whether good or bad) by their middle terms. This was a very clever idea, worth following up on, and, of course, largely ignored. Other authors also expanded Locke's list, so much so that Laurence Sterne takes time to make fun of them in Tristram Shandy, by proposing a new kind of argument, the argumentum ad fistulatorium, which he notes is an almost unanswerable argument: you argue ad fistulatorium when you answer your opponent by whistling a tune.
It was Whately who seems to have made popular the use of the label in discussions of fallacies, although he does not discuss it quite as such. In his Elements of Logic, when talking about ignoratio elenchi (fallacies of irrelevance) he says that "[t]here are certain kinds of argument recounted and named by Logical writers, which we should by no means universally call Fallacies; but which when unfairly used, and so far as they are fallacious, may very well be referred to the present head"; ad verecundiam is one of the ones listed. On Whately's view, ad verecundiam (like ad hominem) can be a perfectly good argument; it only becomes fallacious insofar as you fail to recognize the limitations. Arguments ad verecundiam only establish conclusions insofar as they relate to your opponent; they become fallacious when you take them to establish their conclusion absolutely.
As near as I can tell, however, the treatment of ad verecundiam as fallacious in itself seems to be due to Bentham, who, however, was clear that he was doing something slightly different from Locke. Bentham uses it to help sort out his classification of fallacies whenever he discusses arguments that tend to lead to erroneous conclusions in political contexts (which for Bentham really means the kinds of arguments that people make to argue that we should not accept utilitarian policies). Bentham has some fairly baroque fallacy classifications, but ad verecundiam seems to come up quite a bit. This use seems to predate Whately's usage; Whately, in fact, seems to be trying to get back to a more Lockean usage.
But the history of our mish-mash, crazy-quilt philosophical folklore about fallacies is tricky to follow, so there may be avenues I have not considered.
UPDATE: Whoops! Forgot the link to the article in question. Also fixed some typos.