Cheshires are entirely acorporeal, relative to a human perception. Whether they are made of dark matter or something more exotic, they're not visible or tangible. They have highly complex instincts regarding the matter that they (presumably) use for sustenance, though, and a deep intuition for the rules both of the physical universe, as well as for novel, complex sub-systems such as terrestrial biology.

Using their abilities, if they choose to appear to their prey they can take on any form at all, with most of any texture and composition, though the easiest appearance appears to involve no matter at all, instead simply slashing, fracturing and bending spacetime into shapes, forming the characteristic impossibly silhouette appearance of a Cheshire.

They can procreate both asexually and sexually and lay semi-corporeal eggs, usually into solid corporeal objects such as boulders, rocks, walls, et cetera. If you break open the matter they're encased in, they'll continue to stick to each other in the shape of the now-absent matter, though if they lost their foundation, they would slowly fall. They are very, very light-weight, and look like softly glowing translucent, sticky spheres that contain kitty-spiders.

Much of their attention is done through different senses than the ones that humans are accustomed to. Just because a Cheshire appears to be looking in some direction does not mean it is aware of the things it is looking at in the same way a human would. In fact, a human can use this trait of the Cheshires to 'hide from sight', by taking advantage of the detail that Cheshires sense when visual attention is on them and just not looking at them. That's a fairly good way to disappear - although only if you're not the individual subject of hunt already.

Cheshire society, inasmuch as it can be said to exist at all, is anarchic and individualistic. It appears as if, while Cheshires travel in packs, they tend to instinctively keep almost equidistance toward each other within this pack, leaving every Cheshire with a sizeable territory to indisputably call their own.

Despite their individuality, Cheshires do not differentiate each other with names. They generally don't communicate much in a way humans would consider verbal - and if they do, it's on a level humans can't hear.

Notably, Cheshires don't have a social hierarchy of any sort, either - no Cheshire is particularly in charge of anything. In consequence, they also don't have titles. (You'd be hard pressed getting as much as a job description out of one of them, given their skillsets are practically identical and their knowledge is shared.)

• Dematerialisation; their 'feeding' process, which essentially converts matter into the sort of energy they need to survive. It looks a bit as if they were inhaling the world. (Okay, it looks a lot like they're inhaling the world, if we're perfectly honest.)
  • For victims of the process, it tickles in the first instant and then burns – mostly because you have an open wound, that sort of thing has that effect.
• Since they have no physical form, they can pass through solid objects when they want to.
• Spacetime warping (e.g. teleportation, or 'opening' up an organism without actually killing it, or whatever…).
  • This includes 'singularity sheets' / 'spacetime fault lines', which are semi-permeable space-time membranes. Their permeability depends on the angle of approach you have to them (and with “you” I mean “any matter at all, including photons”) - if you look at them along their normal, they are completely transparent, but they show up at the edge of your vision as a dark sheet. As you pass through (and you should endeavour to pass through quickly, otherwise your blood will disappear as it hits the sheet at a shallow angle), it looks like darkness creeps in on your position, since the angle you're seeing the rest of the sheet at becomes increasingly sharp.
• Less of an ability than a manifestation of their other abilities that crops up occasionally, “Shadow play” is a trick where they extend into their shadow and apply an impossible-seeming force to whatever ends up caught in the shadow to keep it still, e.g. you wouldn't want to get your ankle caught in the shadow of a Cheshire that's engaging in this. They don't do this often, but it's safer to assume they're always doing it.

The most striking weakness of the Cheshires is water in all forms, though the effectiveness of it against them as a deterrent or weapon is entirely dependent on the water's relative purity - meaning not whether it was filtered, but whether or not you would call the resulting object a body of water. The oceans are completely toxic to Cheshires, whereas a human being (consisting to 60%(!) of water), is not toxic to them at all (though the contents of their mouth or stomach might be).

Cheshires can’t pass through pure water – at least certainly not without badly damaging themselves in the process (and if it’s ice, they cannot pass through it at all). Some safety can thus be found in bases under the ocean, but only in reducing the attack surface. Submarines, on the other hand, are essentially impregnable fortresses.

That said, water touching their avatars is only a minor problem. Contact…

• …is intensely uncomfortable (but not painful).
• …interacts with their avatars in a way that destablises them, requiring constant diligence from the Cheshire as not to lose their shape.
• …additionally slowly erodes the avatar on prolonged exposure, much as if acid were stripping it away (but without the associated pyrotechnics, melting or smearing effects - they just gradually diminish, until they are purely acorporeal). This process is not painful, either, just inconvenient.

Unsurprisingly, they hate rain and avoid it.

Pure water is a poison to Cheshires if ingested, but as with all things, the dosage is important. A roughly human-sized body of pure water would kill a Cheshire. Smaller amounts (starting from roughly a mouthful of it) might make the Cheshire rather ill. A poisoned Cheshire’s immune system equivalent is disabled, leaving them open to infections.