It sounds very much like your system lost power before the update was finished. Recovering from such a failure is possible, but most likely not easy.
Since you are not so experienced with Linux my suggestion is to use the livecd boot to copy any file you want to keep, and then reinstall the system. A recent backup will of course make the copy easier. You can copy to memory sticks, other local computers, burn CDroms, use cloud storage or whatever is easiest for you. Remember mail archives and such ...
I don't know how much you've customized your system? You might have account information (eg. mail), netbanking files - but such are usually fairly easy to redo. You might want to take a copy of the entire subtree under /etc - that is where most system configuration is stored.
If you want to try to recover your system anyway, you can try that from the livecd (after copying files of course!). Even better, make a new livecd from 11.04 to make it as close as possible to your system. From the livecd you can use chroot to get a shell "in the installed system" - something like
sudo chroot /tmp
It will start a new shell, which is basically running in your installed system, but with kernel from the livecd (that's why it's best to use an 11.04 livecd. Inside this new shell you can use dpkg*, apt*, aptitude or whatever you use for package management to finish the upgrade. Most likely the upgrade failed while installing files, and the package management system *should* be able to pick up from where it left off. Sorry, I haven't done it all the way myself, so you'll probably need to poke around some in the system and read some man-pages ...
I hope this helped,