How do I tell what version and edition of Windows is on the filesystem?

Look at the file

C:\Windows\System32\License.rtf

Can be found for all these Windows operating systems

Vista, W7, W8, W8.1

These License rtf will have the Version of Windows and the Edition also, Windows 7 Pro for example.

Windows 10 has the rtf file but does not show edition, but if it has the arbitration clause note at the top you know it is W10, or use the explorer trick in Matthew Steeples answer.

To find the Windows 10 product type (Home Pro etc) type see Moif Murphy's answer further down this page

Your best bet is going to be finding the properties of a Windows Executable (such as explorer.exe). In my case it's File and Product version is 6.1.7601.17567

6.0.xxxx.xxxxx will denote Windows Vista (with 6000, 6001 and 6002 being RTM, SP1 and SP2 respectively).

6.1.xxxx.xxxxx will denote Windows 7 (with 7600 and 7601 being RTM and SP1).

The same version numbers will apply for Windows server as well so I'm not sure how you'd tell the difference with those using this method.

Using the registry you can look up version and edition

I found this answer over at Stackoverflow to the question Determine Windows Version, Edition and Service Pack OF AN OFFLINE DISK IMAGE

Use the values under HKLM\SOFTWARE\Microsoft\Windows NT\CurrentVersion. I presume you know where to find that hive?! The respective hive can be found under %SystemRoot%\System32\config with the name SOFTWARE.

Side-note: you can attempt to verify your results by looking at some well-known files (e.g. kernel32.dll, ntdll.dll) and into their version information resource (what you're looking for is the file version: with, e.g. GetFileVersionInfo()).

Edition values, if that's needed, can be found at HKLM\SYSTEM\CurrentControlSet\Control\ProductOptions.

More specifically, how to load the hive of another registry:

http://www.microsoft.com/resources/documentation/windows/xp/all/proddocs/en-us/regedit_load_hive.mspx?mfr=true