Difference between "Browser Mode" and "Document Mode" in Internet Explorer
Solution 1:
Document Mode is what the browser uses to render the page: IE9, IE8, IE7 or Quirks. Browser Mode sets how the browser identifies itself to the web server and to JavaScript.
From a testing standpoint, it seems unnecessarily confusing that these are two separate options and you usually want to change both, for example, set the Browser mode to IE8 and the Document Mode to IE8 to get IE9 to pretend to be IE8.
There's also the Internet Exploer 9 Compatibility View option for Browser Mode. This seems to be the same as IE7 mode except the user-agent string also says "Trident/5.0" (Trident is the IE layout engine).
Solution 2:
In no way this will replace @mhenry1384's answer, but there is also a very good explanation of the differences on the MSDN blogs, more specifically this section:
and here a little more useful info: