NCCA http://www.credentialingexcellence.org/ncca is the agency most associated with accrediting reputable fitness certifications. ACE, ACSM, NSCA, Pilates Method Alliance etc. are all NCCA accreditied certifications. AFFA is accredited through DETC and they post this letter from a lawyer on their site that might be relevent to you http://pdfs.afaa.com/afaa_letter_07-12-11.pdf
I don’t believe Zumba is accredited at all. As far as I know there is no 3rd party oversite of their program.
Many fitness programs are not accredited like Zumba. They may or not be great programs. For example I have the following accredited certifications myself: ACE Group Fitness, NSCA CSCS, and PMA Pilates Instructor. I also am a Stott Pilates Certified Instructor. Stott Pilates is not accredited but their testing and requirements for certification were far more rigorous and harder to attain than any ot the accredited certs I have. On the other hand I could provide you with a Janet fitness certification based on our conversation today that you can hang on your wall and tell all your clients you are a Janet Fitness certified instructor. That would be a meaningless cert. Unless the a program is accredited it is buyer beware whether you are getting a certification of value.