The weave is made up of fibres. Each fibre is actually composed of a certain number of strands. A 1K weave will have 1,000 strands per fibre while a 3K weave will be composed of 3,000 strands. Thus this number can sort of be considered a "density" of carbon strands in a fibre.
For a given modulus of CF, a 3K weave will be three times as heavy as a 1K weave but a structure with a 3K weave will be stiffer than one made with a 1K weave. Note that this does not mean that all 3K structures will be stiffer than all 1K structures as manufacturers will use high modulus CF to achieve in a 1K weave structure what others do using a 3K weave structure.
When considering the material of CF to be used in a structure, an engineer or designer will look at three basic attributes. These are tensile strength, modulus (stiffness) and density. So why would someone choose a lower modulus CF when they can use a higher modulus one, use a lower strand count and still achieve the same stiffness at a lower weight? One word: strength. In actuality, intermediate modulus CF has a much higher tensile strength than high or ultrahigh modulus CF.
Now of course the weave and grade of CF prepreg is just one small part of the equation. The ride quality and strength are chiefly determined by design. A CF structure may not be a homogenous one either. Some layups may have a few layers of high-modulus combined with layers of intermediate modulus. And mixing of portions of the structure of one grade of CF with another section of the same structure with another grade of CF is not uncommon. This process along with directional alignment can be used to tune the properties of the structure.