This varies depending on the organisation.
The title ‘master’ is reserved for higher levels of black belt in Taekwondo. Fourth degree black belts and below are never given the title of ‘master’, and the title normally starts at fifth, sixth, or seventh degree black belt.
In Changheon-yu Taekwondo organisations, the title of ‘master’ is normally awarded to sixth degree black belts and above, as by this point a Taekwondo practitioner will have learnt all of the 24 forms in the Changheon-yu syllabus, and has thus reached the level of mastery.
A large Taekwondo organisation will often have a number of Taekwondo masters in it. Masters are often instructors, and run their own Taekwondo schools, and because of their high status, they will often be involved in the administration of Taekwondo competitions and the Taekwondo organisation in general.
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If you'd like to learn more about the Korean terminology used in Taekwondo, consider buying this book: Taekwondo Terminology. It contains an extensive dictionary of terms used in Taekwondo, as well as explanations on how to pronounce Korean words, and aspects of Korean grammar.