Which Windows 8.1 edition am I running

In the Run window (Win+R) type: cmd /c "slmgr /dli"

You should see something like this:

Name: Windows(R) BLAH edition
Description: Windows(R) Operating System, BLAH channel
...
License status: Licensed

Alternative Solution

You can also use WMI to query the OS information using this command:

wmic os get Caption, Version, OperatingSystemSKU, OSProductSuite

You should see something like this:

Caption                                OperatingSystemSKU  OSProductSuite  Version
Microsoft Windows 8.1 Single Language  100                 768             6.3.9600

OS SKU codes are explained here and here. 100 (0x64) stands for PRODUCT_CORE_SINGLELANGUAGE

OSProductSuite codes are explained here. 768 (0x300) = 512 (0x200) + 256 (0x100)
This means Windows Home Edition is installed and Terminal Services is installed, but only one interactive session is supported.

With PowerShell

(Get-WmiObject -Class Win32_OperatingSystem).Caption

To start PowerShell, hit Win + R (to open Run utility), type in powershell and hit enter.

Open the command line (right click on window, then select Run), type systeminfo.exe and press Enter.

You can use the ver command. There might be some variants I am not aware of. It should print the version of your and / or the version of the command processor.

There are keys in registry in bottom that hold the value of the Ver.