In system tray there usually are two icons related to avast!: a blue round with 'i' letter (Virus Recovery DataBase, VRDB) and another blue round with 'a' letter (the control panel).
To shutdown avast! you have to go through the following procedure:
1) Stop VRDB scanner. Right-click the VRDB icon and choose the corresponding menu item to disable VRDB creation.
2) Stop antivirus scanner. Right-click the control panel icon and choose the menu item to stop access scanner. Confirm your action in the pop-up that will appear.
3) Go to Start -> Control Panel -> Administration -> Services. Stop the services:
“avast! iAVS4 Control Service”
To stop a service right-click at it and choose "Stop".
4) Run Task Manager and kill process ashDisp.exe.
To kill process right-click at it and choose the corresponding item. Confirm your action.