What does the term 'Bootstrap' mean?
In computing, "bootstrapping" refers to software getting to a state to where it can perform the task being requested.

In the context of Drupal, it means the code starts and does everything up to the point where it generates the page.  It more-or-less includes


connecting to the database
reading configuration
loading modules


The bootstrap process is pretty much the same for everything request; what happens when a page gets rendered and sent to the browser differs page-to-page.

So, when "Drupal fails to bootstrap", it means that it wasn't able to get to a point to figure out what to do, and is considered a Fatal Error.  In my experience nine times out of ten when it happens in a production environment, it is because there was a problem connecting to the database.