So, everybody has a "daemon" - it's sort of the series equivalent of a soul. However, it is only in one of the many worlds, these daemons have a physical representation that is in the form of an animal. The daemons can change shapes until the person they're associated with hits puberty, where they pick a final animal form. This is usually somewhat representative of the personality of the person - for example, servants generally have dog daemons, one of the very cunning characters has a monkey, etc.
In most instances, the physical daemons cannot physically go very far from their person, and it is a very close bond. One of the major plot details of the first book is an experimental station where they have discovered how to separate the association between daemon/children, and this is considered extremely cruel (the people are sometimes driven mad, and at best dull and mindless following the procedure). The separation also creates a large amount of energy, and this is used in a variety of different ways throughout the series.