The purpose for certifying hobby rocket motors is to enforce standards on motor manufacturers and to help ensure the safety of the hobby. There are currently four organizations that certify hobby rocket motors. They accept motors certified by the other organizations so launches held under the auspices of any of them allow all certified motors to be flown.
The NAR is the oldest organization and promotes model rocketry in the United States. It has traditionally been focused on model rocketry (A-F class motors), but has started certifying high-power motors in recent years.
TRA was formed to promote high-power rocketry. It is international, with prefectures all over the world. Tripoli is the primary organization for certifying high-power motors manufactured in the United States.
CAR was formed to promote rocketry in Canada and is the primay organization for certifying motors manufactured there, notably by Cesaroni Technology Incorporated.
AMRS was founded by several affiliate rocketry groups already established across Australia. Currently they are only certifying motors for the Australian domestic market with submissions from local manufacturers at their testing facilities.
NFPA 1125 Code for the Manufacture of Model Rocket and High Power Rocket Motors is the base document for the testing requirements for hobby rocket motors in the USA.
The contents of the document is carefully protected against casual reading. (You must purchase it and then even reading it requires access rights on the document.) The license does not allow reproduction here, but there are some interesting tidbits from Chapter 8, Testing and Certification, that are worth paraphrasing (from the 2001 version):
As you can see, there is much lattitude for variation between individual motors. However, most modern manufacturers hold tighter tolerances in production.