Gavindale: I was just looking at the library of games I have purchased on GOG, and for some of my games there is a little blue dot to the left of the game titles. What does this dot mean? At first I thought it was to show what games I have not installed or not yet played, but apparently that is not the case.
To expand upon what others have said: the blue dot shows (sometimes incorrectly) that there has been an update for the indicated game; you can check that game's changelog in your library to see if there were any changes relevant to you (sometimes it's just installer changes, rather than game changes; other times, it might be changes that only affect builds for other operating systems, or the addition of localizations for languages you don't speak; in these cases, you can save yourself a download).
You can filter your library for games that currently have a blue dot by clicking "My collection" (just above the top shelf in your library on the website; not sure how or if you can do it in Galaxy) and selecting "Updated".
Be aware that any game you click on in your library will have both the "NEW" label and the blue "updated" dot removed. Because of this, I would advise that you only click on titles with the update dot when you're prepared to actually download the new files, if necessary -- or just create a custom tag called something like "undownloaded update", "update pending" or something similar, to remind you that your installation and/or backup of a given title is out of date. :)