Detecting 64 bit vs 32 bit

Before starting to look at this, its important to be clear about what you mean by "64 bit".
You can have a 64 bit CPU, a 64 bit operating system and a 64 bit process running.
It is equally possible to have 64 bit CPU, a 32 bit operating system installed and a 16 bit process running.

The table below shows a few of the combinations you may need to account for:

Detect 64 bit processor hardware

There are a number of promising looking options in WMI and Systeminfo, but they all pertain to the OS not the CPU.
Obviously a 64 bit OS must be running 64 bit hardware, but a 32 bit OS could also be running on 64 bit hardware.

Detect a 64 bit Operating System

In Vista and greater, you can use WMIC OS get osarchitecture, or in WMI/CIM Win32_ComputerSystem/OSArchitecture

In batch:

:: Installed OS
Set _os_bitness=64
IF %PROCESSOR_ARCHITECTURE% == x86 (
  IF NOT DEFINED PROCESSOR_ARCHITEW6432 Set _os_bitness=32
  )
Echo Operating System is %_os_bitness% bit

Via David Wang’s blog post: Detect OS Bitness

Windows 10 on ARM includes an x86-on-ARM64 emulation, so the possible values for PROCESSOR_ARCHITECTURE are: AMD64 or IA64 or ARM64 or (for 32 bit) x86

In PowerShell (3.0+) we can use:

[system.environment]::Is64BitOperatingSystem

Detect a 64 bit Process

Running C:\windows\SysWOW64\cmd.exe will launch a 32 bit instance of CMD.exe even if the OS is 64 bit.
Applications that are compiled for a 32 bit processor will run in a similar way.

We can detect this with the PROCESSOR_ARCHITEW6432 variable, AMD64 = a 32 bit process under WOW64 mode:

If "%PROCESSOR_ARCHITEW6432%" == "AMD64" ECHO 32 bit process

Or in PowerShell:
if ($env:PROCESSOR_ARCHITEW6432 -eq "AMD64") {"32 bit process"}

In PowerShell (3.0+) we can use:

[system.environment]::Is64BitProcess

System Folders

64 bit versions of Windows have dynamic system folders C:\Windows\sys* to support both 64 and 32 bit programs.
PowerShell.exe, CMD.exe and many other programs are available as both 32 bit or 64 bit.

The dynamic sys folders will appear differently to a 32 bit session and a 64 bit session:

	32 bit folder	64 bit folder
32 bit session	C:\Windows\system32\	C:\Windows\sysNative\
64 bit session	C:\Windows\sysWOW64\	C:\Windows\system32\

By default a 32 bit session will launch 32bit executables from C:\Windows\System32\ but you can still choose to launch 64 bit executables by specifying SysNative

By default a 64 bit session will launch 64bit executables also from C:\Windows\System32\ but you can still choose to launch 32 bit executables by specifying sysWOW64.

The sysNative folder is not visible to 64 bit processes or programs and cannot been seen in Windows Explorer.

The PROCESSOR_ARCHITEW6432 environment variable. If it’s set, then you’re in a 32-bit process on 64-bit Windows, and must use SysNative. If it’s not set, then you’re either in a 32-bit process on 32-bit Windows, or in a 64-bit process on 64-bit Windows. In that case, you can use System32.

File location environment variables

%ProgramFiles%      = 32 bit programs on 32 bit systems "C:\Program Files"
%ProgramFiles%      = 64 bit programs on 64 bit systems "C:\Program Files"
%ProgramFiles(x86)% = 32 bit programs on 64 bit systems "C:\Program Files (x86)"

Run a 32 bit program

To run a 32 bit program or utility on a 64 bit OS, the 32 bit executable file must be called from C:\Windows\SysWOW64\

if you run a 32 bit shell (such as C:\windows\syswow64\cmd.exe) and then try to launch a command, it will always look for a 32 bit version of the command, even if you explicitly use a full path to system32, the 32 bit shell will redirect to the 32 bit equivalent in syswow64 (if no 32 bit version of the command is found, then the new process will fail to launch.)

Run a 64 bit program from a 32 bit process

To run a 64 bit program from a 32 bit process use the virtual folder C:\Windows\sysnative

In many cases this is not needed as most utilities (e.g. ping.exe) have both a 32 bit and 64 bit version, however a few utilities (nbtstat, bcdedit) are only available as a 64-bit executable.

By default, running CMD from the start menu will launch a 64 bit process (C:\Windows\System32\cmd.exe)

“It's not so much that we're afraid of change or so in love with the old ways, but it's that place in between that we fear... it's like being between trapezes” ~ Marilyn Ferguson

Related:

Q556009 - How to check if a computer is running a 32 bit or 64 bit Operating System.
Q896458 - 64-bit Windows does not support 16-bit applications.
WMIC OS Get OSArchitecture