Windows 8: How Can I Tell If I'm Running 32-Bit or 64-Bit?

The easiest way is to click "Desktop" (at the bottom left on the image above);

Windows 8: Start Page

Then when desktop appears bring up Windows Explorer;

Windows 8: Desktop

Right-click "Computer" (on the left) and select Properties;

Windows 8: Getting Machine Properties

This then brings up a dialog with information about your installation;

Windows 8: System Properties Dialog

I've highlighted the bit which tells you whether you're running 32 or 64-bit version of Windows.