Most CPUs manufactured by AMD and Intel over the past four years are 64-bit compatible, but if you plan to upgrade it's worth checking. You can easily do this online: if your CPU supports either ‘AMD64' or ‘Intel 64', it's compatible with 64-bit Windows.
You can also check from within Windows Vista: in the Control Panel, go to the Performance Information and Tools display. Click View and print detailed performance and system information. You'll see a technical breakdown of your system that details, among other things, whether your processor is 64-bit capable.
A second factor to consider is compatibility with hardware devices. 32-bit device drivers won't work in 64-bit Windows, so if you want to keep your existing peripherals and components you'll need to find 64-bit-specific drivers. That shouldn't be a problem for devices manufactured over the past few years, but for older hardware check the manufacturer's website.
Remember that Windows 7 can normally use Vista drivers - so even if a particular gadget lacks drivers specific to the new OS, it may still be usable.
Another piece of good news is that you don't need to worry too much about application compatibility. Almost all modern 32-bit software should install and run flawlessly on a 64-bit edition of Windows.
The only exceptions you're likely to encounter are programs that try to hook into the system at a low level - such as 32-bit antivirus software or plugins for video-editing packages - and the odd badly written hobbyist program.
It's also worth noting that 16-bit apps (such as old DOS games) aren't supported in 64-bit Windows, but there are third-party tools, such as the free DOSBox x86 emulator, to help keep very old software running.
Also in this series, 64-bit Windows:
Part 1: Should you switch?