If you download FireFox and want to install it, you simply copy the app from the mounted diskimage to your /Application directory. That won't ask for your password if you've logged in as an admin user, as you have the "right" to copy items to that folder within the Finder.
A password is needed if the installer writes to directories where the admin user doesn't have write privileges. In that case the installer does a sort-of "sudo" to temporarily gain root access to be able to do the install action.
In other cases, the installer of an app will ask for an admin password simply because it is programmed to do so, even if you have all the privileges. And IMHO I think all installers should do that.
One of the small irritations I have with the way Mac OS X handles software (non)-installations, is that non-admin users can download an app, like FireFox, and copy the app to their Desktop (or even leave it in the mounted diskimage..) and execute it.
IMHO developers shouldn't distribute their app like that.