A Juicy Lucy is a cheeseburger with the cheese inside the patty instead of on top. That's it. That's the whole idea, and two bars on the south side of Minneapolis have been fighting over it since the 1950s.
Matt's Bar spells it "Jucy Lucy" and will tell you they invented it. The 5-8 Club spells it correctly and will also tell you they invented it. Nobody can prove anything. Everyone has a side. I have been to both more times than my doctor would like.
The one rule that matters: wait. The cheese inside comes out at roughly the temperature of the sun. Every Minnesotan has burned the roof of their mouth on their first one, and every Minnesotan watches visitors do it anyway, because it's tradition.
My side? Matt's, for the fries in a paper bag and the grill you can see from your booth. Fight me in the comments.