What is the Leica look that fans of the company claim is unique to them? What concrete differences are there that are most obvious to a layman?
I think there were a number of real reasons for 'a look', and most of those no longer apply...but others may have taken their place.
The real ones IMHO are historic.
35mm Changed Photography
Leica invented 35mm and suddenly a lot of professional reportage and suchlike became infinitely more portable. This meant that there were Leica type photographs, that had nothing to do with the quality of the image.
35mm Changed Image Quality
'Actual' image quality actually took a nosedive with 35mm, but so long as this was factored...it could be said it was not 'worse' but 'different'.
Grain increased but portability meant reduced use of lights and more pushed ASA/ISOs for motion shots. The result for street or war photography was a texture. Meanwhile where film speed was not so critical, 35mm was still sufficient to capture fairly grain-free images. We should remember analogue crystal grain can be attractive in a way that digital noise never shall.
At the time Leica was making its name in the '40s and '50s as the only 35mm pro system, I understand monochrome media used far higher concentrations of silver nitrate than later film. Images from that time (the actual prints, not scans on a monitor) have a luminance and quality that I think was lost in later film. As such, by the time all the other pro 35mm systems came out they could not match the old Leica images ..the 'Leica look'...but here's the crunch...nor now could Leica!
The Olympus OM-1 was perhaps the next game changer, it was actively designed to knock Leica off its pedestal and largely succeeded in professional circles. It actually fits the Leica case, up to but of course not including, the pentaprism and was originally called the M1 until Leica objected. It too had uncoated lenses and again IMHO matches the 'Leica look' allowing for the fact the silver nitrate was now reduced for all.
So what then is the latter day 35mm Leica look? A high-performing professional uncoated lens, but I disagree that this has to be particularly sharp. What it has to do is have a very good bokeh, good contrast, and a fast aperture.
Most Leica owners seem to stick to a prime. In contrast, most SLR owners these days tend to use a zoom. Zooms are slower than primes, so you lose the narrow depth of field the prime offers and they generally render less well.
Also the zoom means many fail to really get up close to their subject...so this can make the image appear flatter and relatively lifeless. IMHO it was a sad day Olympus added zooms to their OM system.
Uncoated lenses were superior for monochrome but had issues for colour. Leica tends to go with monochrome a lot.
I tend to use uncoated lenses on my DSLR when shooting monochrome and even on digital I think it does make a difference, but like the notion that a modern Leica has an inherent look, I might be kidding myself! It definitely makes a vast difference shooting infrared. Most modern lenses, irrespective of quality, are plagued with issues on an infrared DSLR.
I think a modern photograph on a Leica has no more of a special look than any other professional camera with a professional prime in the same kind of price band. Any suggestion that it is superior is almost certainly just that many compare the images with those taken on comparatively budget kit and usually shot with a zoom. Likewise anyone investing in a Leica would have to be pretty serious about their photography and probably fairly experienced...so the typical Leica shot is not really comparable with the typical shot of a consumer camera.
A FORMAT COMPARISON
Today we see micro four thirds creating pro solutions on tiny sensors with many pros rejecting the pro claim because of this. Noise increases, resolution diminishes, "negative" size is limited at around 16mpx and one needs twice as fast lenses to get compsrably shallow depth of field, just as was the case moving from medium or large format to 35mm, if we can claim that was ever achieved.
As such, within MFT, "Acme Camera Co." charged the Earth but made only top notch super fast lenses and made super compact portable bodies by exploiting the small formar..they'd be a Leica scenario for today. Pretty soon people would talk of an "Acme look".
Personally I find the Leica M series actually feel bulky, even " vlunky" today. I used an M2 for a while and have to say it never came close to the joy of the Olympus fitted with an f/1.2 lens, though the rangefinder was incredibly bright ( unlike on a series lll or its Russian clones, and worked a treat.
Insofar as the Internet has decided this...
The Leica look is a low depth-of field image, with sections that are out of focus. Point light sources turn into well-formed circles, and other out-of-focus forms are modified by a similar blur effect. A quick Google Image search for "leica bokeh" is illustrative here.
In terms of stereotype, it doesn't hurt if it's a high-contrast black-and-white candid street photo, showing off the film's dynamic range and alluding to the camera's history in the "street photography" genre. :P
Leica makes high-quality lenses with good capabilities (fast, low-distortion, etc) ... but so do plenty of other people. To some extent the Leica look is just a little bit of showing off with a good lens. But the configuration of the glass and aperture that produces the blur in out-of-focus areas is the hardest to replicate.
Just to give another (maybe wrong) view on this, I think what is being referred to as the Leica look is a mixture of subjective and objective things.
There are some photographers that are so used to review, make, create, and study pictures taken with Leica lenses and cameras that can identify the lens a picture was taken with. I know it may seem disturbing, but there are people that work every day with photos, and, if they happen to have a bias for Leica, they take great attention and study of these things (i.e. the subtle differences of images produced by Leica). This may be seen as the subjective part.
The objective part is that there are relatively few Leica lenses, and of course they share much in terms of design, designer, and glass type used, so inevitably they share some of character in terms of color reproduction and bokeh. Whatever is the Leica look, it is produced by that set of lenses.
Hope to help here.
I think different people will have different definitions, but the so called Leica look has been often associated with the look produced by lenses created by Walter Mandler. No lens is perfect and each designer needs to decide their priorities and set of compromises they are willing to make. Mandler selected the compromises in a way that differed from mainstream manufacturers and achieved a distinct look, also known as "Leica glow".
There are a lot of non-technical things about the 'Leica look': choice of subject matter, composition and so on. To a great extent you can achieve these things with any modern camera, although some aspects are easier with something a little more discrete and quiet than a giant (D)SLR. However it's important to realise that, historically, it was possible to make pictures with a Leica that were extremely hard before it: there are things you can do with a camera you can carry discretely in one hand that are quite hard to do with a camera which you use on a tripod, especially in a world where everyone expected a camera to live on a tripod, and these things were an important component of the 'Leica look' because the Leica was the first camera that let you do that. Interestingly, today, a Leica is one of the least discrete cameras you can carry: no-one ever comments on my Pentax MX, but a lot of people comment on my Leica!
There are also some technical aspects of the Leica look: 35mm film meant relatively grainy photos and still does mean that. Black and white means, well, B/W. A lot of classic pictures made with a Leica were made with Tri-X (but not the modern formulation of Tri-X) which is a fairly fast film so has even more grain and looks like, well, Tri-X.
And perhaps the most important technical aspect: the lenses. I think the term here is 'Leica glow': this is something you get with a lens which was optically-good by the standards of lenses designed before computers and which is uncoated or at most single-coated: it will have relatively low contrast (good for B/W) and be prone to various sorts of flare aka 'glow'. It will have good (or, anyway, not-offensive) bokeh. It will be a prime (optically-good zooms were rare before computer design, and you effectively can't use a zoom with a rangefinder). It will not be very long or very short because, again, rangefinder: say 28mm to 90mm. And this in turn reflects back on the kind of pictures that you take, of course.
The Leica Look: In my opinion only applies to Film cameras, where the final product (Fiber based or RC coated photographic print with an enlarger) is processed with a Negative developed and used by a Leica Camera (Range finder M Leica) The effect that you will see and difficult to describe is the Leica look. In the digital realm (Digital Leica) there are too many variables between the shooting of a photograph, the photoshop processing and final result print that it is difficult to compare and define.