ACSM stands for American College of Sports Medicine and, as a personal trainer, certification through them is known as the gold standard of certification. It requires more work and it costs more money, but the likelihood of getting a well paying personal training job and being respected when starting a business ist very high if you have one of the three categories of ACSM certifications.
AFAA stands for Aerobics and Fitness Association of America. AFAA personal trainer certification is a fast way to get certification, and the entire certification process can be done completely online.
The National Council on Strength and Fitness (NCSF) is an agency accredited through the National Commission for Certifying Agencies (NCCA). They offer certifications for personal trainers.