The fallacy that the notion X is true, only because authority Y says so. No matter who the authority is, God, Prophet, holy book, the President, or Mr. John Smith; the authority still needs a reason to believe the notion, and that reason has to be clear. Sometimes, the reason is perfectly clear. A doctor has a specialty and a definite insight into the illnesses s/he has specialized in. It is perfectly reasonable to assume a specialist has good reasons for her claims. Therefore it won’t be fallacious to back a claim about one’s health and cite the doctor as the source of it.
To make this more clear, we could put it this way: Authority cannot be replaced as a premise of an argument, or as the reason behind a claim. However, a legitimate authority can be put as the source of reasons behind a claim.
Example: There is quite an interesting issues on morality which could be pointed out under this fallacy. A lot of moral claims given by fundamentalists turn out to be purely fallacious, on both fronts: replacing the authority with reason, or appealing to an authority which by no means is even remotely close to being a specialist on the subject of moral claims. Sometimes it is even worse, the authority turns out to be completely devoid of any sense of morality.
One of these particular issues is the law itself, when the law is presented as the only reason for the correctness of a notion. In a back and forth conversation with some pro-guns after the shooting in Sandy Hook elementary school (in 2012), they kept pointing out that “We have the right to have guns, our constitution is clear about it.” and I kept asking them “It’s true that the law in US allows people to have weapons, but why do you think it’s the right thing to do? Why the law is right? What is the reason?”
The law does not make a notion automatically right simply because it is “the law”. There are reasons behind what our politicians decide to legislate, and simply pointing at a certain law does not make a similar claim right.
Perhaps the worst of all appeals to authority are the claims from religious fundamentalists on the subject of moral values. In arguments with religious fanatics, “X is wrong” is a notion that is usually backed up by “Because God has commanded it”. Obviously God (any God) is by no means a legitimate authority on moral subjects. Most of it could be because God never seems to clearly answer questions about his reasons for a particular commandment. And moreover, by reading most religious books we immediately realize that most Gods are worse than psychopaths, how could they ever be a legitimate authority on moral subjects?
 Compare this with the case of legitimate authority who specializes on a subject. For example a scientist on his/her specialty A legitimate authority is always prepared to provide reasons for what s/he believes.