How can I tell if a Windows 10 PC has the Anniversary Update installed?

Press WinKey + R, type winver and press ENTER

If the version number reads 1607, then you have the Anniversary Update installed.

You can determine this by looking at the Start menu, since the Anniversary Update changed it. Before the update, it looked like this (with words):

After the update, it looks like this (just icons):

A particularly salient difference is that the user avatar/name no longer appear at the top; rather, the avatar is above the File Explorer icon.