CFA - Cat Fanciers Association
TICA - The International Cat Association
ACFA - American Cat Fanciers Association
CFF - Cat Fanciers Federation
CFA is the oldest and best known association. Most people register with CFA and one or more of the other associations. Not every association accepts every kind of breed but the standards for each breed are very close in description so you are able to put your cat in a show in any of them and it should be able to compete.
Many people show in several associations and win titles in each. You can't transfer titles from one to the other but if your cat is good enough they can win multiple titles. For example, my 14 yr old cornish rex is a Grand Premier in CFA, a Quad Grand Champion in ACFA and the one TICA show he was in, he qualified as a Grand but I never registered him at the time. He is also a multi time national and regional winner in ACFA (one year was 10th best alter nationally!)
BTW, ACFA is NOT a smaller association - its the 3rd largest in the US (TICA is a spin off from ACFA)