It really sounds like it is booting a partially installed copy from the hard drive. Check the boot order in the BIOS and make sure the CD is still first...
The strange thing is, I tried again later last night and it booted just fine. So I unmounted the HDD, did a format, and remounted it. I figured that should take care of any stray OS that made it onto the hard drive. Rebooted again and... it asked for my login. So unless the format didn't actually work, it seems like it's asking for the login from the CD.
Tried rebooting a few more times, and it seems to be randomly either booting up normally, asking for a login, or just hanging at the point where it'd normally pop up asking if you want to run off CD or install. The last time it did the latter, I powered down and happened to notice a bunch of "authentication failure" messages as it was going down.
So on a whim, I threw in a Fedora CD and tried booting to that, and during the boot it threw an error about I/O problems.
At this point, my guess is that it's having trouble reading either the hard disk, or the CD drive, and that Ubuntu is (intermittently) interpreting that as needing authentication. I'm not sure how/why that would be the case, but it's a guess. I may just throw in another hard drive and see if that changes anything.
It's also interesting that it died half way through the install, to me that suggests that either the CD is corrupt (and there should be an option at CD boot to check the integrity of the files on it) or the machine itself is a bit unstable. Make sure it isn't overclocked, check for bulging caps and check that all the drive cables are plugged in securely. If you are still having problems when you get it booting off the CD properly and installing again then run a memtest on it. Ubuntu install failures either happen at the beginning if it has problems booting on the hardware or at the end when it goes to reboot off the newly installed disk, not in the middle of copying files.
I thought it might be a bad CD so I tried burning a new one, with 10.10 (originally I was installing 10.4 LTS) but the machine doesn't seem to like 10.10 - it throws an error during the boot saying it can't mount the volume. Will try burning a new 10.4 copy tonight.
The machine has never been overclocked or anything like that. Cables all seemed fine, but I'll remove and re-seat them tonight.