bun. - Chewy? Crusty? Moldy? My IDEAL bun is of the pretzel variety, but since that never really comes around I usually stick to a hefty but chewy bun. Maybe a bit of a crunch. I can be persuaded.
meat-ness. - Honestly, the actual burger of a burger takes backseat when I'm driving, but I still like my beef (or chicken, or pork, ...or, hell, tofu, I don't care really) to be...good. You know. (That being said, I also love places that offer veg and vegan options for all my friends.)
toppings. - Now, here's where it gets interesting. I like a lot of different things on burgers, but mostly I like how they work together. Also you need to offer caramelized onions as a topping or you are not a true burger establishment.
sauce. - This is where it gets real important. I don't like my burgers without sauce. Doesn't have to be super crazy or anything, but sauce is what brings this lovely burger together. And the hint here is: home-made barbeque.
uniqueness. - I don't usually go that out of the box when building my own burger, so I like when joints do the thinking for me! Examples: eggplant as a topping, diced chicken instead of an actual burger, and bacon cooked in pancake batter. You really can have it all.
overall gnarliness. - Not to sound like a twenty year old ripped surfer dude, but I usually describe a good burger as "gnarly." They sort of attack the plate, attack your mouth, leave you gasping for air and all...basically like that crazy push of a wave I rode this morning, man. tubular.