I've read the book and can't find a definition of "initiative". I'm talking about the weapons which list "initiative" values from +2 to -6, that I've seen so far, but I'm not real sure how that impacts the weapon's actions. Is it a matter of speed, or number of attacks?

It seems to affect both, but it plays a direct role in combat initiative.. when during the round the wielder will get to swing, shoot, bash, etc.
In a less direct role,it looks like a large penalty item can reduce the number of swings you get each attack, and vice-versa for small quick weapons.