Yes you can have them installed on the same computer, but you must have already partitioned your HDD to have 2 seperate HDDs.
If you try to install on the smae Partition, the older one will get deleted.
If you already have it running and not have another partition like what was mentioned by cristobal, you will have to either format and lose all your data, or you can insert another HDD and install it there.
Yes you properly have to reload everything like what the other guys have said.
Oh but take note though, how old is your old computer?
I heard from a friend that his old PIII refused to boot on RH9 without a keyboard...