This password to protect the avast settings isn't set by default, so either someone with access to the system has set one, or perhaps at some point this option Set/Change Password... might have been clicked, see image.
Backing out of it might have caused it to have a null or space in the value, if you click OK without entering anything that might resolve it. However, if a password has been set and you don't know it, effectively your only option is to reinstall avast.
It may be best to do a clean install:
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. Ensure that you scroll down and select the avast direct download link for the English version and not Cnet as that is for an on-line installation (not what you want to do). - Direct download for avast Home, English version,
http://files.avast.com/iavs4pro/setupeng.exe. I assume it is the Home version, if not get the latest Pro trial version.
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.