The book is in twelve chapters, each given the name of a month of the year (starting, not with January as you might expect, but with August). Du Sautoy tells of his life as a mathematician, describing some of the problems he works on, and the experience of solving such a problem after having battled with it for months or years - but then there's always the worry that there's a hidden flaw. We also find out about his holidays, which also involve plenty of mathematics - on his visit to the Alhambra palace in Granada, du Sautoy was determined to find examples of all 17 two dimensional symmetries in the wall patterns.
A significant thread in the book is the history of symmetry, from the Platonic solids, through the development of algebra up to the work of Galois, and continuing with the development of group theory in the 19th and 20th century. The reader is thus introduced to the 'Monster' group, and its relationships with totally a separate part of mathematics - the 'Moonshine' of the title (In the USA the book is entitled Symmetry: A Journey into the Patterns of Nature though.)
It's an entertaining read, and I'd recommend it to anyone who wants a glimpse into the life of abstract mathematics with a minimum of technicalities.