This "should" work: open a terminal window and type "uname -m".
A 64-bit system will report "x86_64" (or maybe "amd64" -- I didn't
burn a 64-bit Live CD yet); a 32-bit system will report "i686" or
"i386" or something similar.

I have two Ubuntu machines, a desktop that has a
32-bit CPU and a laptop that has a dual-core 64-bit CPU. The command
"uname -a" gives "Linux jim-desktop 2.6.27-7-generic #1 SMP Thu Oct 30
04:18:38 UTC 2008 i686 GNU/Linux" for the desktop and the same info for
the laptop (except for jim-laptop).

Since they both say the same thing, must be that they are both running a
32-bit OS.

I just booted the 8.10 amd64 CD I burned and "uname -m" does display
"x86_64" which confirms a 64-bit system.

Until just now I wasn't sure if it'd return "x86_64" or "amd64" since
the ISO file is named ubuntu-8.10-desktop-amd64.iso

You can find the general info using the GUI at System | About Ubuntu. However I
don't know if it tells you whether 64 0r 32 bit and does not give specific
kernel info.

In the terminal you can type: uname - a (to find the kernel version)
and : lsb_release -a (to find which version of Ubuntu)
or: cat /etc/issue

I hope this is what you want.

None tell me whether 32 or 64 bit. Looks
like I am running a 32 bit OS (see my answer to
Thad a few minutes ago).

My 64bit Ubuntu shows in grub as I have both installed with something like
Ubuntu 8.10 kernel 2.6.27x on sda10 and Ubuntu 64 8.10 kernel 2.6.27x on sda 11.
I know both from the partition as I always keep a list of installed distros (I
have 11 or 12) or 64 is explicitly mentioned in grub whereas 32-bit does not
mention this. The same can be said for MEPIS 64 or Fedora 64. MEPIS says 64
right after the kernel (2.6.27-smp 64-bit) and Fedora says Fedora 64 before the
kernel and location. This is from grub as written by Ubuntu 8.10 but I get
similar results when grub is from MEPIS or similar. Fedora seems to ignore most
installed distros so I don't like their implementation much. I am doing this
from memory because I don't want to reboot, but it is generally correct.