The monks are grouped according to how often they have contact with the outside world, which can be every 1, 10, 100 or 1000 years. As the story starts, it's just before a 10 year Apert, where the 1 and 10 year monks will mingle with the populace for 10 days. Erasmus, a "tenner", finds the monastery's astronomical observatory closed, and gets the first hints that the secular and monastic authorities are conspiring to keep a pretty big secret. Together with his cohort of young monks, he gets drawn into solving the mystery.
Saunt Descartes was a drunken fart
In the early part of the book, Stephenson draws the reader into the world of Erasmus's monastery. He uses the common SF trick of making up words for things: the monks are the "avout", the outsiders "saecular", the monasteries are "concents". Some people don't like this sort of thing, but for the most part I was content to let this wash over me as part of the book's scene setting, the measured pace of which parallels the life of the avout. (There's also a glossary at the back, which helps). The avout are serious seekers of knowledge who learn the stories of theoreticians as religious monks might learn about the lives of saints (the avout word for a great thinker is "saunt", a contraction of "savant"). They engage in debates which are intellectual duels, the sort of stuff you get in the better debating places on and off-line. Stephenson has placed real philosophy in the book under the names of the saunts who thought of it on Arbre: it's fun to try to work out the real world analogues, among whom are Plato, Faraday, Occam, and Einstein. The philosophy isn't just there for show, it becomes important later: Stephenson is the second SF author I've come across who has written a story which hinges on the idea of the Platonic world of forms (the other is Greg Egan).
Modern life is rubbish
In comparison with the concents, the saecular world Erasmus encounters in the 10 days of Apert is unthinkingly religious and commercialised, a parody of modern American society right down to the thugs in sportswear (anyone who remembers the thetes in The Diamond Age might think that Stephenson has a thing about this) and the drugs which keep everyone happy but somehow blunted. Erasmus observes several times that clever people tend to end up inside the walls of the concents. The book seems to describe a vicious circle of anti-intellectualism leading to the intellectuals hiding away, leading to further distrust of intellectualism in the outside world, which eventually leads to the concents being sacked every few thousand years. The initial retreat into concents happened because of some cataclysmic events in the past. You can see Stephenson drawing on A Canticle for Leibowitz here, with the difference that the avout aren't just preserving old books they don't understand.
The first part of the story is a more erudite version of a Harry Potter book, with the young avout (Erasmus is 18 as the story starts) ranging over the old stone buildings they live in, talking about philosophy and science, and finding ways around authority with the help of some wiser older monks. We see more of the saecular world as Erasmus is thrust into it in the later part of the book, and finds that things aren't a total cultural desert out there. Stephenson dislikes the unthinkingly religious and so Erasmus does too, but the religious contemplatives that Erasmus meets show the other side of Stephenson's opinions, where religion provides people with a code which keeps them from the feckless behaviour of most people outside the concents.
Ninja monks in space
The final part of the book is page-turning SF stuff with ninja monks in space, a long way from Erasmus's quiet life as the book begins. Stephenson draws the philosophical threads from earlier in the book into a satisfying conclusion. The popular notion that he can't write endings was disproved by The System of the World, but sceptics will be pleased to hear that Anathem has an ending too.
A positive effect of the narrator's voice is that the book is less frenetically digressive than Stephenson's earlier stuff. Some of Stephenson's wild tangents are fun (my favourite is the wisdom tooth removal in Cryptonomicon), but they make his books longer without advancing the plot. At about 900 pages, Anathem is long, but most of it is world-building or action (if you count the philosophy stuff we're going to need for the later revelations as "world-building"). Other reviewers have complained it's slow to get going, but the avout are sympathetic characters, so I didn't mind reading about their lives at the start of the book. I think it'd be quite cool to be one of them, in fact.
Anathem is a fun mix of philosophy and action. Recommended to people who read the sort of stuff I write here on LJ :-)