How to run multiple user X sessions on the same computer, at the same time?

If you want multiple users to use Keyboards/Mouse then that is multiseatX.See this documentation

With first X session running login as user you want to use and run in an xterm or at a console prompt:

xinit -- :1

or

startx -- :1

If you are using Gnome:

Try running gdmflexiserver, it will launch a new GDM session. You can switch between them using Ctrl+Alt+F{8..12}.

Take note though that logging in with the same user twice is a bad idea.

Also, (in Lucid, at least) you can switch users by clicking the Power button in top-right corner and then selecting a user name.