Windows 7 uses the same kernel as Windows Vista. The kernel is the core of the OS that runs everything.
So as Dragon said, if it runs on Vista, it should run on Windows 7 just fine. I'm running 7 and Vista on various machines, and frankly 7 isn't that much different, aside from some GUI enhancements clearly designed to take on MacOS. Vista had quite a few bugs at first but if you keep up with your updates and therefore have the latest service pack for Vista, there isn't a whole lot of reason to upgrade.
An upgrade from XP however might be worth it.
One thing that Windows 7 has is a more robust virtualization feature set, with the intent of being able to run something called "XP Mode." I haven't used that particular feature myself, but basically it allows you to run XP within 7 so if you have any old applications that do not work with anything newer than XP, you can still run them with 7. You could do this with Vista as well, but it wasn't nearly as integrated and required a lot more techie knowledge.
You can learn about XP mode here:
http://www.microsoft.com/windows/virtual-pc/