It will run just fine by itself. However, even with iCloud, mine says it'll need to use a comptuer to do software upgrades. Also, unless you purchase all your content from iTunes, the computer is needed to get data and music and movies etc onto the iPad.
So it will run just fine by itself, but still needs a computer every once in a while to transfer data and do software updates.
I did my last update to iOS 5.0.1 over the air via WiFi so you should not need the computer to update the software on the iPad.
If you are running iOS 5 you do not need a computer - but if you buy movies from iTunes and you do not transfer them to your iTunes library on your computer - you cannot download them again for free if you restore and lose the movies.
You don't actually need the computer to use an iPad - but personally - I would never recommend it. Downloading everything that you could lose if something goes wrong on the iPad and you have to restore it - one app at a time from iCloud - would be a real pain in the butt IMO.