In, say, Windows Server 2008 -assuming the installation wasn't under the Server Core mode-, and in general, will applications made for the general purpose OS line (XP/Vista/7) run?
I mean programs like audio/video editors, videogames, IDEs...
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It's the identical OS to Windows 7, the only thing you'll find is that the aero effects don't look quite as crisp, but it should handle everything else just fine. Bear in mind it can be used for terminal services, i.e. it can be used for user desktops, so 99.999% of productivity and other software that runs on windows 7 can run on server 2008, usually software that doesn't will be restricted only by the developer.
Because Server 2008 is designed to run as a Terminal Server, nearly everything can if not everything. What few don't run out of the box generally start working once you install the Desktop Experience feature.
Where you'll run into problems is not with Windows directly, it'll be with 3rd party applications. Some applications will explicitly not run on Server 2008 because it hasn't been 'certified'. Others require specific drivers that are not tested against Server 2008 and may not install.