If you have a mixed breed cat you can still show him/her. They have a special class called "Household Pet"
The cats really don't do anything, like in a dog show. You take them to each ring, the judge will put them on a small table and examine them for health, grooming, personality. In purebreds you have a written standard that the cat is judged by.
Basically you want a cat that is not afraid to be handled by strangers, likes to play with toys and is used to crowds. Some cats hate it, some tolerate it, and some really enjoy it.
You can look up local shows in your area in Cat Fanciers Association, The International Cat Association, American Cat Fanciers Assoc. and a few smaller ones (in the United States). Every weekend there is a show or several going on across the country. Maybe you can find one in your area to go see what they do.
The expense varies - the entry fee may average about $45-50 depending on the number of rings - usually 6-12 rings. So if you break it down per ring, you are only paying about $5 per ring or less.