If you aren't logged in with the local admin account and you do something that requires that authorisation, then you will be prompted for the password or UAC.
This is why it is advised not to use just the local admin account should someone gain control of your computer - but I don't bother, I just have the single built in admin account and not password protected.
Because I'm only using the single admin account without a password, I get straight to my desktop on boot and when I want to install anything, I'm presented with the UAC which I just have to click on Yes.
Delete those two accounts using your admin account, remove the password and set up your accounts again without passwords.