GDM is the "Gnome Display Manager"
it is the programm you should see,
if you want to use a graphical login
(so you type in your name and pw, then it starts Gnome/Kde/XFce4 for you)
The GDM is the most widely used display manager,
but others like kdm, xdm ... existst also.
Because X needs high priveliges to run,
the gdm developers were extra cautious to who has
access to the gdm config and runtime files.
One of these sanity checks gdm runs at startup is to check if it is the owner
of its own directotries (/var/gdm)
Normaly you should issue a
"chown gdm.gdm /var/gdm" as root
to reset the correct ownership.
The question you should ask yourself is
"How did this happen?",
these kind of permissions don't get reset during normal operation,
but only through fideling........