Appeal to authority is commonly heard as the name of a fallacy. However, sometimes it is appropriate to appeal to an authority, as when we have questions about a topic which is generally relegated to experts, and where the experts are in relatively good agreement. If you want to know how far away the Earth is from the Sun, appeal to authority is fine, and just about the only way most of us will learn that information. Similarly, germ theory, basic genetic theory, and the top ten hits are all things we know, most of us (if we know them), on the basis of authority. The question, then, is when appeal to authority is fallacious.
The most obvious case where appeal to authority is fallacious--and this is covered in any decent logic textbook--is when the authority is not an authority in the right field. Stephen Hawking on religion, or, the usual example, pick a celebrity on almost any topic.
The other case where an appeal to authority is insufficient is when someone else can establish the contrary by appeal to an equally valid authority. This may be regarded as a subset of the above, but where no one can claim to be an authority on the issue. There are an enormous variety of cases where we depend on this kind of authority, however. Medical professions have many areas where we must follow some authority or another, but they disagree and are equally valid authorities. How are we to proceed? Here the concept of a burden of proof is helpful. The way we tend to go is to follow the established majority, which may or may not be right. We cannot all study the issues involved, so some kind of authority must be involved, but the authorities, on certain issues, disagree--and this is true in any scientific field, the issue is that in medicine we must go on.
So there is a puzzle here. We cannot simply appeal to anyone with a medical degree as an authority, but we cannot all get the proficiency with concepts required to evaluate the research ourselves--and if we did, we would likely have a smaller chance of getting at the truth than those whose profession it is. So, in general, we go with the majority unless we have encountered arguments and evidence which provide reason to think that the majority is suspect. Such arguments can be very difficult to evaluate, however, because they can easily use technical jargon which appears to establish a claim, yet which might not.
With issues like these it is helpful to step back and establish heuristics which we can apply to the conclusions, of the form "if the studies give x kind of conclusion, something has probably gone wrong." For example, if studies seem to show that we should, in general, eat in a radically different way than we have always eaten, either the reason should be shown in how we live radically differently (and there is probably an issue with how we are now living in that case) or something has probably gone quite wrong. Another might be, for a Christian, if studies show we should altogether avoid things which the Israelites were supposed to eat (e.g., meat, unleavened bread, wine), then something has likely gone wrong--which is not to say that we ought to eat the same as the Israelites did. Also, the right diet is liable to result in our enjoying food more. And those are just about diet. Things regarding surgery and other medication would be harder to establish, but hard thought could probably get some kind of conclusion-heuristic thinking running sufficiently to enable us to evaluate at some level the more disagreed upon practices. Some of that thinking goes on in bioethics, but this kind of thinking would be broader than that.
This kind of approach needs to be done carefully. We need to avoid gut-reaction kind of thinking that says "I don't think God would allow such-and-such" unless we have good reasons, grounded in Scripture, for thinking so. I have heard people who don't believe wormholes or time-travel are possible for this reason, which strikes me as odd. Our God is big enough to handle all sorts of things which might make us uncomfortable, and humans have, historically, been uncomfortable with all sorts of things. People once claimed that flight was impossible because if God had wanted us to fly he would have given us wings. These are cautionary tales of this kind of conclusion-heuristic thinking going wrong.
The final case that I intend to cover here is where the question is such that the idea of an authority on the issue is itself invalid. Again, this can be construed as a form of the first or second instances, where we are all counted as authorities and yet many reasonable people disagree. What is an example of this? In my view, there are some philosophical and theological issues which fall into this category. Even granted that Scripture is always a valid authority, the issues we address in theological and philosophical discourse usually cannot be settled by an appeal to another mere human's argument or view, even if that other person has a PhD in a relevant field. This does not mean that we all need to delve into such issues at the PhD level, and it does not mean that PhD's are no better off than the rest of us, but it does mean that if we are to claim a doctrine we should understand the arguments pro and con at some level, at least long enough to see how the arguments work. With theology, because it is always applied, we need to have enough of a grasp of where the doctrines come from to understand how they should work out, and to establish a great enough confidence to live them out. The conclusion-heuristic thinking approach can be helpful here, too. One very important criteria is this: does this doctrine either rest on or support the essential gospel of doctrines? That is, does the doctrine either show the cross in a more beautiful light or under gird the cross work of Jesus Christ? True doctrines tend to wind up at the cross at some point. True doctrines should serve to increase love for God and others.
Life under God is communal, and thus we have teachers in the Church, but it is also individual, and so we each must come to know and love God for ourselves.