The Challenge is a vital part of Garou culture, and is a primary method of social stratification. It is the Master of the Challenge who hears and arbitrates all Challenge requests at Moots and otherwise. Serving as something of a Marshall of the Lists, she organizes Challenges so that there is no overlap.
From Guardians of the Caerns, pp.73-74:
"The Master of the Challenge is necessary to making sure that the sept's members can cooperate smoothly. By overseeing all challenges, whether grudge matches or tests for the Rite of Accomplishment, the Master of the Challenge reinforces the sept's hierarchy; she's responsible for ensuring that all Garou feel they're being treated fairly.
The Master of the Challenge should ideally be eminently fair, but in practice she tends to reflect the attitudes and values of the sept. The Master of the Challenge of a Fenrir sept is going to be harsher and less forgiving than her counterpart in a Stargazer sept; she'll probably make sure that each challenge makes both participants stronger no matter what /they/ want. Most Garou who fill this post are Philodox, but that's a general tendency and not a prejudice. A comprehensive knowledge of gamecraft is a definite requirement; plenty of combat experience is also necessary. The Master of the Challenge need not be unbeatable in all forms of challenges (and who /could/ be?), but she must be good enough at all of them to be well versed in picking the appropriate challenge for a situation."
For information about challenging, see the news file on challenging.
Former Masters of the Challenge at the Hidden Walk (list is incomplete):