This is a rope that is constructed from different layers of strands where the layers' torsional stiffness balances each other out. Non rotating ropes are most often steel wire ropes, but there is no reason why this cannot also apply to polyester ropes or even hemp ropes.
When creating a non-rotating rope, the layers in the rope are laid in different directions. For example you can have the inner layer with right-hand lay and a layer of left-hand lay on the outside counterbalancing it. The result of this is that when you put load on the wire, it will not twist and elongate due to the rotation.
This kind of rope is also called a rotation resistant wire rope.
Right and left hand lay is used to describe in which direction the rope is spun. Right hand is clockwise, and left hand is in the opposite direction.
As you can see from the sketch of the cross section of the rope (not to scale) the non-rotating wire relies on smaller strands of wire, and is much more compact in it's construction. You can barely glimpse the two layers of which the non-rotating wire is built up from.
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\o/ o o \o/ oo xx xx xx oo
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Rotating wire Non-rotating wire
( Single layer with core ( Right hand lay on outside
right-hand lay ) layer, while inside layer
and core is left hand lay )
Why is this important ?
The reason for using non-rotating wire rope is that when you put force on the rope, it will elongate and rotate. This rotation will spin the object you are lifting, causing you to lose control of the object. This is not that important for light loads that you can push into place - but for precision positioning of heavier loads this becomes a critical factor.
A rotating load is also a safety concern - especially if it is unsymmetric. Sharp edges don't really matter as the rotation speed is so fast that any impact will at least break some bones.
Most "at home" ropes are rotating, so an easy way of checking this is to take a bucket of water and tie a rope onto it's handle. If the bucket twists when you lift it, then you have a rotating rope. You will get a larger result by using more weight in the bucket and more rope when lifting.