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.
The Athletic Certification Training, or ACT, Commission offers a free personal trainer certification through their website. Traditionally, the certification costs about $300, but in order to streamline the process and introduce their website to people who are not familiar with it, they are now offering their certification free of charge.
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.