Generally speaking, Appeal to Authority is simply saying that something is true based on the fact of someone else saying so. For example, "The world can't be millions of years old, because the bible says it's only a few thousand years old."
Science itself, however, would grind to a halt without the ability to rely on what others say.
The scientific method doesn't rely on statements like "X says Y, therefore Y is true". It is more like "If Y, as X says, is true, then my observation of Z implies my hypothesis regarding A is true."
The argument then can be falsified by showing X was wrong about Y or it can be weakened by showing that X did not say Y.
This is in fact the true basis for nearly every cdesign proponentist objection to science. "Sure, all you guys claim to have shown that this evilushun stuff is true, but I've never seen it. And my authority says it ain't so." And, while the cdesign proponentist's claim for support for his position is certainly vulnerable to attack based on it being an Appeal to Authority, the interesting thing is that so is the science position.
cdesign proponentist's arguments can be show false in the same way. Usually because ultimately X did not say Y, but sometimes because X has been subsequently shown to be wrong. Sometimes the error in logic comes in the claim that their hypothesis is implied to be true as a result of Y.
Also, cdesign proponentist's often appeal to authorities who are not actually authorities in the field under question, they are often dentists, urologists and engineers making statements about genetics and the like.
In the final analysis, science is really nothing more than a popularity contest. When the number of people who claim to have confirmed a given proposition reaches a certain critical mass, the proposition is generally accepted.
It is a contest of ideas, the ideas with the most merit and evidence become more and more accepted. When many experts independently agree with a proposition, non-experts can be assured that the proposition represents the best knowledge available. Tentativity, as always, is present making sure the argument is never "The consensus of experts of field X says Y and Y is a proposition under the field X, therefore Y is true", but rather "The consensus of experts of field X says Y and Y is a proposition under the field X, therefore Y is the best proposition we have. Given Y, then....".
If two historians who specialis in 11th Century warfare say that such a Battle occurred in 1065, then that is more likely to be true than any contradictory statement by an economist and a carpenter. All human knowledge builds off standing on the shoulders of giants, not just science, and it is not fallacious to do so. As long as it is always borne in mind that the experts should continually test the foundations, the shoulders, for their reliability. Otherwise dogma and orthodoxy rear their ugly heads.
Edited by Modulous, : No reason given.