Welcome to Android Central! Are you referring to different sessions of Facebook? The easiest thing to do is to log in on Facebook on your desktop or laptop. Click on the downward arrowhead at the upper right corner of the screen, then click Settings. Select Security, then click on Active Sessions. You can now look at all of your active Facebook sessions, and end any of them that you don't recognize by clicking End Activity.
Not sure why you're getting a notification on your phone that your Facebook login failed. Are you able to login to the Facebook app without any problem? If so, then it might be some other app on your phone that also tries to access Facebook. Check if there are any preinstalled apps from Motorola like a Social Hub or something like that, whose purpose is to consolidate all of your social networks like Facebook or Twitter into one place. An app like that might be trying to log in with an out-of-date password.
Actually, just found this support page describing the Droid X's Social Network function, which is like what I just described above. That's probably what is causing the login failure.
https://motorola-global-portal.custhelp.com/app/answers/detail/a_id/48488