A good set of tools to inspect and debug webpages is the “Internet Explorer Developer Tools”. Although I use them regularly, I have never used the browser and document mode until very recently (probably because I did not have a good understanding of what browser and document mode meant).
In one of the links below I found the following explanation:
While the Browser Mode affects how Internet Explorer reports version information to servers and webpages, the Document Mode affects the document compatibility mode reported by the Web page to Internet Explorer.
So in other words with Browser mode a user / developer can test how a website looks like in another browser. Here we say to the web browser: “Render the content as if you were another browser”.
However using Document mode one – most likely a developer – can test which is the best document mode to use in his webpage. The document mode gives information to the web browser about how it should be rendered. Here the webpage states to the web browser: “Please render me like this”. In this case the browser version remains the same.
Discovering Internet Explorer Developer Tools
How to use F12 Developer Tools to Debug your Webpages
Testing Browser and Document Compatibility Modes with the Developer Tools
How IE8 Determines Document Mode
Quirks mode (wikipedia)