A Messal is a dining group, by tradition at the Concent of Tredeghar. (Read in the context of the novel, it is likely a contraction of the term "mess hall.") Each Messal has seven diners, and seven other servitors who prepare and serve the meal. The seven who dine are known as doyns and are typically more senior Avout, while the servitors will be more junior. Each one has a particular doyn they serve. The number seven is chosen as the largest group which will maintain a cohesive conversation among all participants, without breaking-off into subgroups.
The servitors gain by the chance to listen to the conversations of more experienced Avout. They are generally not to speak unless invited, however those who are respected will from time to time be called upon. Later, when they become more senior they will get the chance to be doyns. It is a tradition among servitors that they should leave the dining room and go to the kitchen if they do not like the topic being discussed. Arsibalt believes doyns are supposed to notice this.
In Anathem we primarily see the very special Plurality of Worlds Messal. This is special for any number of reasons, including the presence of a Millenarian and a Saecular Power official, as well as of course, Jules Verne Durand.
The concept of a Messal is foreign to those from the Concent of Saunt Edhar, where a more egalitarian refectory system is used.