x32 means 32-bit OS, or 32-bit processor. There are still some of those around.
x64 means 64-bit OS or processor.
x86 means Intel 8086 family processor (286,386,486, Pentium etc.) as opposed to PowerPC, ARM, Sparc etc.
A 64-bit OS will not run on 32-bit hardware
A 32-bit OS will probably run on current 64-bit hardware because it includes compatibility mode instructions.
A 64-bit program will not run on a 32-bit OS
A 32-bit program will probably run on a 64-bit OS, though it might need 32-bit compatibility libraries. In Linux, they are not always loaded by default.