How to write a good MathSciNet review? When reviewing for MathSciNet, I routinely find myself just paraphrasing and abbreviating the introduction provided by the author, and occasionally adding a few words about the quality of the research or the cleverness of the argument (which the authors themselves would not be able to write for obvious reasons). There seems to be very little added value in doing this, since the paper already has the introduction (which has the added benefit of being written by someone who has intimate knowledge of the paper) as well as abstract (which will in most cases be sufficient to decide if the paper is worth reading). Of course, I can imagine edge cases when someone can't quite decide if the paper is worth delving into based on the abstract alone, while the introduction is for some reason difficult to read or the paper is difficult to access. But I can't help feeling that there should be more to it.
I would love to hear opinions about what makes a MathSciNet review useful, and how to achieve it.
Edit to add: As YCor correctly points out, the same question applies with ZBMath or any other place that hosts public reviews in place of MathSciNet. To avoid creating a question which is a moving target, I will refrain from making edits above.
 A: I'll take a stab at this because in the past I have gotten some feedback from Mathematical Reviews saying that they like my reviews, and they did ask me to write a Featured Review once (back when there were such things as Featured Reviews).
The answer to the question depends to some extent on how long a review you want to write.  My default length is probably around two to three times the length of the abstract.  I typically try to at least give a precise statement of the main result(s).  Often the abstract does not do this because stating the main result requires quite a bit of notation and preliminary definitions, which are too long to put in the abstract, but which usually can fit into a review.  I do this because I imagine that some people might have access to MathSciNet but for some reason don't have access to the paper, and a precise theorem statement might help them decide whether to put in the extra effort to obtain the paper itself.
Another thing I do is to put myself in the shoes of a MathSciNet user, who is looking for relevant papers that he or she is not currently aware of.  I ask myself, what keywords can I put into the review that will help such people discover this paper via a keyword search?  If you look through the paper with this mindset, you will often find remarks about related topics that will provide good keywords.  These don't always make it into the abstract or the introduction, but it's useful to have them in the review.  If you happen to know that the objects in the paper are sometimes studied under a different name then that's also something useful to mention in the review.
If you're willing to put in the effort, MR will happily accept longer reviews.  As I understand it, the Featured Reviews that MR used to have were discontinued for various reasons (e.g., I heard that, contrary to MR's intent, Featured Reviews were being used by the community for hiring and promotion decisions, and MR did not feel qualified to decide which papers were the "best"), but there is nothing to stop you from writing something similar for any paper you feel like.  You can search for "featured review" in the review text of reviews from June 2005 or earlier to get a feeling for what these were.  Well-written Featured Reviews were not only longer and more detailed than the typical review, they were written with a wide audience in mind.  The idea was that a Featured Review would convey some idea of the context and significance of the paper to non-specialists.  I will freely admit that I rarely have the energy to write such reviews, but they are certainly of value.  Imagine someone stumbling upon your review in their search results and finding your review more accessible than the paper itself; they could very well make a conceptual connection that they wouldn't have otherwise, or be drawn into an area that is close to their own interests but that they didn't know existed.
A: I'm the Executive Editor at Mathematical Reviews.  I'm impressed by some of the answers, particularly those from Timothy Chow and Denis Serre, which give as good an answer as I could hope for. As Timothy Chow and Joe Silverman point out, having links to other items in MathSciNet related to the current paper is especially helpful to users.  Kimball quotes the salient bits of our Guide for Reviewers.  Thank you for doing that. People tend to forget that the Guide exists.  When I talk with people in person at a conference, I compress the goal of a good review down to: describe something about the context of the result (where did it come from?); describe the main result (what is it?); describe the main technique(s) or method(s) (how did they do it?).  Doing all this concisely can be difficult, of course.
The AMS Bulletin still publishes reprints of reviews.  They are generally tied to one of the articles in the issue.  While they are generally above average, they are not necessarily meant to be the best of the best.  For a time, I was posting particularly good reviews that I came across.  You can find them via this link: exceptional reviews. I should probably get back to posting examples of good reviews.  I second Yemon Choi's recommendation for Kimball's collection of (his version of) exceptional reviews.
A: You say that your reviews are "paraphrasing and abbreviating the introduction." That's incredibly useful, since most introductions are multiple pages long, so having the key information available in a couple of paragraphs is exactly what people need to know if the paper is relevant and something that they want to read further. Thus I can skim a dozen Math Reviews about the research area I'm studying in the time it would take to skim the introduction of just one or two of those articles. Indeed, it takes time and knowledge to do a good job of "paraphrasing and abbreviating" technical material, which provides significant added value to the author's introduction. And many (most?) author abstracts are not as clear or useful as a well-written math review.
One other aspect of a review on MathSciNet that I find very useful (and try to include in my own reviews) are cross-references to the most relevant associated papers. Yes, those papers are usually included in the list of references, but the list of references tends to be long, and it's very helpful to have two or three especially relevant cross-references to click on and see their reviews. So in addition to the explicit content of the review, this is a significant added value to the abstract that the author(s) included in their paper.
A: To write a review is, to some extent, a journalistic endeavour- you are reporting on the result, presenting it to  an audience that is wider than the researcher for which the paper was perhaps initially intended. As such, I think that what I can add in a review is a wider perspective than authors may present in either in the abstract or in the introduction.
First, I may be receiving a paper to review several years after publication, and subsequent developments may be relevant to somebody who wants to decide whether to read the paper. There may also be parallel or independent work which the author(s) may not have been aware of when they submitted the manuscript.
Second, papers will commonly present their results to a narrow audience, and I may be able to introduce those results to a wider audience.
Third, some papers are written by people for who have difficulty writing prose in English (and a good introduction may be prose as much as it is mathematics), and I might be able to explain the result more clearly in terms of English prose than what appears in the abstract and in the introduction.
A: I am not any more a MathSciNet reviewer, but I have been so for about three decades. I am happy to share my experience.
It is true that a review is often a paraphrase of the introduction, if not of the abstract. The abstract is even explicitly given in place of the review.
This was not my taste. I departed from the abstract or the introduction when I had a special motivation. This happened to me many times, but I remember very well the review that I wrote about the book How fluids unmix, by J. Levelt Sengers. When I started reading it, I could not stop before the end, and I wrote a very long review, expecting to motivate people to have a look at it.
Of course I have been asked several times to write a Featured review. But on one occasion, I realized that the papers made an unmotivated assumption, and that the resulting analysis missed the key point. I told MSN that a featured review was not appropriate. I wrote a neutral review and then did the analysis that the topic deserved (I confess that this was borderline, on the professional level, and I should like to have the opinion of other participants).
My third contribution is that I had to fight several times against a reviewer who did not believe in continuum mechanics (!). This man wrote extremely negative reviews about every paper dealing with continuum models. Many prominent mathematicians have been his victim (I was not personally, but just by chance). This reviewer departed too from the traditional form, but in the wrong way.
A: While several users have given useful advice, I encourage you to read the
Mathematical Reviews Guide for Reviewers
to get their opinion.  It starts by telling you what a review is (supposed to be). In particular, it says

A review should primarily help the reader decide whether or not to read the original item. The review may range in length from a few lines to about 600 words. In most cases the review should state the main results, together with sufficient information to make them comprehensible to someone already familiar with the field. The main ideas of the proof should be sketched when feasible. If the results are technical, requiring extensive notation or elaborate formulas, it is preferable to describe them with a few well-chosen and relatively nontechnical sentences. A review should also contain comments that provide some background for the item, evaluate it and connect it to related items or approaches.

And under "Evaluative reviews"

Your review may include a positive or negative evaluation of the item. Critical remarks should be objective, precise, documented and expressed in good taste. Vague criticism offends authors and fails to enlighten the reader. If you conclude that the item duplicates earlier work, you must cite specific references. If you believe there is an error in the item, please describe it precisely in your review and provide evidence validating your claim (e.g., a counterexample, an exact reference which supports your assertion, or an indication where the error arises in the paper).

I'd also suggested reading a bunch of existing reviews, to get your own sense of what is good and what isn't.
