The full details of the mechanics of Combat Initiative are not fully understood. For detailed research, please see the spading work and the Initiative Guide being developed on the Hardcore Oxygenation Forums -- the discussion of mechanics is drawn directly from there.
This is a comprehensive list of everything in the game that directly affects Combat Initiative. Higher initiative increases the frequency with which you are the first to attack when encountering monsters, and improves your chances of being able to flee successfully from a combat (according to Jick in this post). To see the maximum levels of initiative available to players, see Maximizing Your Combat Initiative.