An IETF-defined protocol for administering and securing remote access to a network.
RADIUS server software includes three parts: an authentication server, client protocols and an accounting server. These pieces can all run on one machine or on separate ones outfitted with different operating systems. It works by having a user dial in to a remote access server and pass logon name and password information to it. The information is forwarded to a RADIUS authentication server that validates the user and returns the information necessary for the access server to initiate a session with the user.
A dictionary file kept in the RADIUS database determines the types of attributes that can be included in the user profile. The user repeats this process to initiate every session.
Additional resources
RFC 2865
IETF RFC that discusses and defines RADIUS.
A little on the history of RADIUS
Here's a background on the history of RADIUS: History of the RADIUS Server