On the 4th of July, we Americans don't merely celebrate our country's independence from the British Empire—we also pay our humble respects to that most American of comfort foods: the cheeseburger. But what types of cheeses work best? As I see it there are three variables to consider: meltability, tanginess, and the funk factor. Some choose to optimize for one of these variables, but there may be a perfect cheese to satisfy all three.
If you're going to go for cheeses that melt perfectly, processed cheese is the way to go. Whether it's Velveeta or Kraft American Singles, processed cheeses make up for their bland flavor with undeniably superior meltability. (They contain added emulsifiers which help prevent the separation of water, fats, and proteins when heat is applied.) Processed cheeses are also the way to go if you want to really showcase the flavor of the meat—the cheese will add a nice texture to the sandwich without interfering taste-wise. However, there are other meltable cheeses that potentially have much more flavor: Gruyère, Comté, Brie, Taleggio, Fontina, and many more.
If you're willing to forgo some of the smoothness of the cheeses listed above, and would rather have your cheese add a sharp bite to the sandwich, your best bet is to go for an aged Cheddar, Monterey Jack, aged Provolone, or even Parmigiano-Reggiano. While some of these cheeses may separate a bit when hot, they are also much more flavorful than your average processed cheese. Tangy cheeses also tend to balance the savoriness of the meat really well.
The Funk Factor
If you really want a cheese that will contribute a great deal of flavor to the overall cheeseburger experience, go for a blue. It is best to choose a blue that is fairly rugged so that you don't end up with a saucy mess. Think Maytag Blue rather than Gorgonzola. If you go the blue route, don't be upset if all you taste is the cheese—blue cheese is super strong and will often overpower the subtle flavors of the meat. But if funkiness is your thing, a blue cheeseburger can be a revelatory experience.
Ultimately I think that a really good aged Gruyère or Comté is the best way to go if you want to maximize all three factors. These Swiss/French Alpine cheeses melt really well (they are the fondue cheeses after all), they have a little bit of tang, and if you find the right specimen they can add that nice funky edge as well.
What do you look for in your cheeseburger cheese?