Each logical avman server is composed of two parts: a dabase back-end process and a server process proper. These normally reside on the same physical machine but in cases of very busy avmans, MakenaTechnologies have split an avman over two physical machines with the database back-end process separated from the server process (source: forum post).
An avman is responsible for managing non-spatial data such as user accounts and object ownership, commerce transactions and ThereBucks balances, clubs and groups, events and so forth. Note that object possession and object ownership are distinct: possession is tracked by the 3D servers (a dob knows who is carrying it) while ownership is tracked by the avman (the pob for a dob knows who owns the dob).
The avman is usually to blame for effects like slow login, startup inventory issues and auctions timing out.
When avman3 died in April 2004 and had to be restored from backup, one effect was that all objects designed by one of the avatars assigned to avman3 vanished from the PortaZones they had been placed in.