The installation mechanism by default requires a previous operating system to be installed, but as the directions linked by Rich above indicate, an alternative method bypasses that requirement. However, legally speaking you need to own a qualifying operating system to upgrade.
The good thing is that the upgrade applies to more than one operating system. According to Microsoft, "With an upgrade copy, you can install Windows Vista if you already own a compliant, licensed version of Windows 2000, Windows XP, or Windows Vista." I believe this applies to Windows 7 as well (with the exception of Windows 2000).
So, if you have an old copy of Windows XP still lying around, you can use that as the surrogate base operating system and "upgrade" it. That would still leave you able to legally use your existing copy of Windows Vista on the current computer. This is very useful for those who wish to have a dual boot system.
The only exception to this is OEM versions. An OEM version is an operating system that is pre-installed on your computer. It is non-transferable and cannot be used in another computer. So, if that computer "bites the dust" so does the operating system.