if I set this value to x86 does that mean I cannot run that project on a x64 machine?
No, 32-bit applications (x86) run just fine on 64-bit Windows (x64). All 64-bit versions of Windows include a 32-bit compatibility layer called Windows on Windows 64 (WOW64). This is usually what you want, in fact, as most applications do not benefit from being complied for 64-bit.
However, compiling for 64-bit (x64) does mean that your app will not run on a 32-bit (x86) machine. You can go backwards (64-bit can run 32-bit), but you cannot go forwards (32-bit cannot run 64-bit).
Compiling for Any CPU is always an option, as you point out. That will allow the application to run as a 32-bit application (x86) on a 32-bit machine, and as a 64-bit application (x64) on a 64-bit machine. This sounds like a panacea, but there are costs. Most notably, you’ll need to test your application extensively in both 32-bit and 64-bit environments, whereas if you only target 32-bit environments (including 32-bit environments on a 64-bit host), you only have to test one build. And the additional workload is rarely worth it—most business applications do not benefit from the extra memory space of a 64-bit environment, and probably end up defeating any potential gains by the increased overhead of 64-bit pointers.
Visual Studio itself is a good example of an application that is fully 32-bit. There is no 64-bit version, yet it runs fine on a 64-bit host. This blog post helps to shed some light on why the decision has been made to keep VS 32-bit. You might find the reasoning helpful in making the decision yourself.
Likewise, although Microsoft Office is now available in a 64-bit package, Microsoft is still recommending that most customers stick with the 32-bit version. There are compatibility problems with the 64-bit version, and there just isn’t much benefit.