Although they should never be used as an argument in and of themselves, analogies are very useful forms of illustration, simplifying a given topic by way of comparison to something similar. The main problem with analogies is that, by definition, they are not synonyms. No matter how many similarities two analogous things share, there will always be at least one difference.
The false analogy fallacy occurs when the similarities are too superficial, or the differences are too significant, for the analogy to be valid. In short, it is when the comparison just doesn't make sense.
An obvious example of this is when atheists compare the Bible to Harry Potter. The similarities are superficial. Yes, they are both books, as the atheists are quite quick to remind us. However, this is where the similarities end. The fundamental purpose is different, the content is different, the authors are different, the evidence is different.
Regarding the fundamental purpose, no one is intended to believe Harry Potter is based on real events that happened to real people. The Bible is written with the intent that it be believed.
Regarding the content, the Bible is chock full of all sorts. Songs of praise, historical narrative, moral guidance, doctrine, rebuke, correction, all with the intention of bringing us into a reconciled relationship with our Creator. Harry Potter is just a relatively straightforward narrative with no practical application, or real world implications.
Regarding the authors, one woman wrote Harry Potter, and it came primarily from the inventions of her own mind. She did not see the events. She did not hear of the events from anyone who did. She did not gather official Hogwarts records and condense them in a useful way. The Bible, by contrast, is a long collection of records, a lot of which is from eyewitnesses and contemporaries. Of course, the ultimate author is God Himself, but as this is the very point being disputed by those who would make such a false analogy, we would be as guilty of circular reasoning to assert this as they are to deny it.
Regarding the evidence, the Bible is the most well-verified book in all of human history. There are, of course, significant gaps in a 2,000 year old book that covers 4,000 years of human history. However, based on what we do have left, there is so much evidence, it takes effort to deny it. We have fulfilled prophecy, we have confirmed locations, we have internal consistency, we even have scientific confirmation (including minor anachronisms). But find me even one shred of evidence for Harry Potter.
By nature of analogy, there will always be some differences. A difference alone does not make an analogy a false one. That is why, for example, Evolutionists cannot refute the mouse trap analogy by pointing out that mouse traps cannot reproduce. The mouse trap analogy is designed to illustrate irreducible complexity, a concept Darwin himself recognised as a threat to his religion. The addition of reproduction to mouse traps would only amplify the problem. Thus, the similarities between man-made designs like mouse traps, and God-made designs, like the mice themselves, are not superficial. Rather, it is the difference, in this analogy, that is inconsequential.
When the similarities are more significant than the differences, the false analogy has not occurred. However, when a similarity is overblown, or a significant difference is not duly acknowledged, the false analogy fallacy has occurred. The counter to this fallacy is as simple as exposing this. Highlight the significant difference, or reject the superficial comparison.