A Cutaway Gag is a comic element found in some comedy television shows. The sequence generally opens up with character saying a completely random line, then the scene immediately cuts to the action. The scene usually lasts a few seconds before returning to the main plot line. These scenes generally don't have anything to do with the story.
In Family Guy, Cutaway Gags are commonly used in most episodes. They are used after a sentence or conversation, usually starting with "This is even worse/better than the time..." and are the main comedy part of Family Guy.
It is implied that these gags are simply stories made up by the people that set up the gag, and that the incidents shown did not really happen. This is particularly obvious when it comes to parodies of movies and television shows, or incidents that show a character dying.