If you are from the US, I can give you the basics.
In all the associations (CFA, ACFA, TICA) you have a Household Pet Class for non-pedigree cats. These can be mixed breed or purebred without papers. If over the age of 8 months, they must be neutered or spayed to compete. CFA bans any declawed cat.
They are basically judged on grooming, health (muscle tone, etc.) and then on personality. HHP's are pretty competitive - just like the purebreds. ACFA and TICA register HHP's and give out year end awards to the top winning cats; same as if they were purebred. They can earn titles too equal to a champion or grand champion.
Most people start out with showing in the HHP class and then move on to purebreds. And most purebred showers have shown in the HHP at one time or another
Its a lot of fun and you learn a lot. The judges usually love judging this class cause there is no written standard