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.