What is the cat show judge judging?
Every breed of cat which is recognized by the CFA has a written “breed standard” which describes all the characteristics which would make for the “perfect” version of that breed.
The judges are not, technically, judging the cats against each other, but against that “ideal cat” described by the standard.
Standards obviously vary for each breed, but they usually mention such things as the overall size of the cat, its pattern and color, ear shape and “set” (where they sit on the cat’s head) body shape (long and thin, cobby, squarish, rectangular…) length and quality of coat, eye color and position, muzzle shape and other attributes.
You will often see judges using a teaser toy of some sort to get a cat to play or climb up a pole. While they are interested in seeing cats who are happy and engaged, they are mostly using the toy to help them see certain features of the cat, such as its length or the profile of its head or its chin shape.