It shouldn't keep asking you for the password and I have no Idea why it is, a corruption of a setting perhaps, I simply don't know. I have had avast for almost five years I have never set a password and it has never asked me for one and as you have said up until now neither have you.
Why are you trying to turn avast off (and how) ?
avast 4.8 has a self-defence module and that would pop-up so are you sure it isn't that, an image might help ?
I know very little about Vista but I do know many swear at the UAC, but from what you say I don't believe that it might be a limited user account thing as the UAC would normally only ask when starting an application.
The only area of avast with a password is that for protecting program settings, etc. Does anyone else have access to the computer ?
If you updated to Vista (I don't know what your previous OS was) it may be necessary to reinstall avast and if you a) didn't set a password, b) don't know what that might be, then the only option is to reinstall.
It would probably be best to do a clean reinstall, first Download the latest version of avast
http://www.avast.com/eng/download-avast-home.html and save it to your HDD, somewhere you can find it again. Use that when you reinstall.
Download the avast! Uninstall Utility,
find it here and save it to your HDD.
1. Now uninstall (using add remove programs, if you can't do that start from the next step), reboot.
2. run the avast! Uninstall Utility, reboot. If step 1 failed it may be necessary to run this from safe mode, once complete reboot into normal mode.
3. install the latest version, reboot.