First of all, please rememberl ... every judge can be different in how s/he judges. Each is apt to have his/her pet peeve and will pounce on that one thing, more than another! And there'll be some things a judge may never even look at. How much they scrutinize, may depend on how stiff (quantity and quality) the competition is
What do they look for?
* binding, straight and even
* pucker free
* corners of quilt square, and then the binding corner mitred perfectly and square with that corner
* inside of binding is "full" (when you machine sew on the binding, and flip it to the other side, make sure it is pulled all the way over, so that the binding is full with the batting)
* bias joins, including the final join
* hand stitching next thing to invisible
I'm sure there's more, but this is what first comes to mind!
It's a fair question to ask if the Fair will provide the judging standards card to you in advance of the show. Some will, some won't.
And it's not just the binding that is going to win or lose for your entry. Attention to detail throughout the construction and finish and making sure that all is straight, square and precise is going to take you a long way to the finish line!
Good Luck! and have FUN with your entries!