For me, a big part of what makes someone move well is coordination, or working on a particular skill set. So gymnasts have all these skills that they are required to learn, such as acrobatics on the floor and the vault, this kind of supported work on the horse and pbars, and this swinging work on the high bars (did I miss any? oh wait - trampoline too!). Most of what gymnasts do is very high level coordination, spinning multiple times in the air, coordinating huge weight shifts while supported, etc. Circus performers have similar practices with trapese and hand balancing and their various stunts and tricks, martial artists have very complicated forms, parkour people have crazy skills for moving through the environment.
As I have mentioned in a previous post, I kind of strive to be this "expert generalist", someone with the best and most practical all-round ability. In terms of conditioning, Coach Sommer's material is by far the best out there for this, with the greatest boost to overall ability compared to other conditioning programs I've tried. I am also reminded that gymnastics is traditionally a practice where the skills came first and conditioning was this kind of "built in" thing that just came as a result of practicing skills, however Coach found out with his athletes that instead of this 90/10 split of skill/conditioning, that somewhere closer to 50/50 produced much better results. That being said, Coach obviously still has his guys practicing their skills a lot and doesn't focus purely on conditioning.
From a practical standpoint, I find the parkour/movnat idea of being able to move skillfully through the environment very useful in day to day life, so the majority of my "skill" practice is this. I also use my martial arts to practice some lower body physical elements and my internal skills (once again very useful in a number of practical situations, in particular rest & recovery, and of course in fighting).
What does your skill work involve and why?