When I currently reboot, I just get the option of the user I want to log in as. It has me and "other" which asks me for the user name.
The only user name I can think of is root, but I don't know its password. Am I doing something wrong? How do I get myself into the administrative boot console from which I can get into the single user mode?
Thanks, Alex
You're waiting too long during boot time. The login screen you're referring to is part of your (most likely) Gnome desktop manager.
When you are booting up, there should a period of time when you're screen turns black or a small cursor flashes at the top of the screen. Usually hitting escape will give you the grub screen, as on most distros this screen is suppressed by default.
Since you're on Ubuntu, see https://help.ubuntu.com/community/GrubHowto