turn radius is a numerical measure of how tight a curve you can make in a turn on your skis to give you an idea of how the ski is meant to be ridden, and comes about due to the difference in sidecut of the tip and tail to under foot. so with small turn radius skis (10m-20m) the ski is going to want to make tight arced turns, and conversely a ski with a large turn radius (generally 30+m) will want to make large radius arcing turns like the huge wide turns you see when big mountain riders come into lower, less technical wider sections of a line. a small turn radius doesnt mean you cant make a wide arcing turn it will just be more squirrelly and hooky than on a ski with a larger turn radius, and vice versa. in general park skis have turn radii on the lower end of the scale as the spaces you are riding in aren't really wide and you need plenty of maneuverabilty on the snow to zip between features, all mountain skis generally have middle of the range turn radii to give a bit of both the characteristics of small and large turn radii. Traditionally serious big mountain skis were in the larger radii scale to be able to do the afore mentioned huge arcing turns. as serious pow skis have become wider, its more about float than being able to arc a turn with the effective edge of your skis as you just cant in pow.there is another concept called dual turn radius which has 2 separate turn radii built into the sidecut of the ski allowing short turns to be made when you put more pressure on the front of your skis, usually in steep and/or technical sections and a wider turn radius when you are putting more pressure on the rear of your ski, like when you are on a more comfortable, less inclined slope and have more space to turn in.
this dual turn radius is more of an all mountain ski feature but its not used exclusively on all mountain skis.
Of late there have been powder skis like the JJ which have very short turn radii due to a distinct sidecut on the short sections of flat cambered or cambered parts of skis to give the skis some versatility. In my opinion this is a total waste of a serious powder ski, as giving a ski sidecut reduces float and the only reason a ski has sidecut is to allow it to turn on harder more compact snow, which is counter to the proposed use of the ski. sure its not always going to be bottomless pow your riding on and you want to be able to ride it back to the lift after ripping the shit out of a pow line but a short turn radius is not necessary to do that.
Here marks the end of my lecture on turn radius and short rant about how bad the JJs are imo.