Whether you run a restaurant that serves Mexican-inspired cuisine, or you just like cooking Mexican food at home, it is important to use authentic ingredients. This also applies to any cheese you might use. While Monterey Jack and cheddar cheeses are delicious in their own right, using them in Mexican cuisine is not terribly authentic. If you really want to up the ante, then use one of these real Mexican cheeses instead.
Oaxaca is a cheese named after its place of origin. It looks a lot like mozzarella in that it is white, stringy, and comes in a ball. It also melts easily. However Oaxaca has a bit more body and flavor than mozzarella, and it's not quite as stringy when melted. It works really well inside quesadillas or melted on top of tostadas.
Requeson is most similar to ricotta cheese. It is spreadable and soft, and it's generally used as a filling for empanadas. Reuqeson does not have a lot of flavor itself, but it is excellent at taking on the flavor of any seasonings and sauces you might combine with it. This cheese stands up well to baking and long/slow cooking methods.
Asadero comes from the state of Chihuahua in Mexico. It's a semi-soft cheese with a texture similar to that of Havarti. Asadero cheese melts easily, but it's also delicious when crumbled or grated on top of salads and other cold dishes. Its creaminess helps calm down spice from hot peppers.
The name "queso fresco" translates to mean "fresh cheese," and indeed, this cheese is fresh. It's a non-aged cheese made from raw cow's milk. As a result, its flavor is incredibly milky. Some versions are pressed so gently that you can still separate the individual curds. Queso fresco is often used to stuff tortillas, or as a topping for spicy dishes.
Queso anejo is an aged, goat-milk cheese. It is very firm and has a hint of spice since it is traditionally rolled in paprika during the aging process. This crumbly cheese is salt and pungent. A little goes a long way. It can be sprinkled on top of dishes as a seasoning or finishing touch.
Each of these Mexican cheeses is unique, but they'll all add something special to your food. Pick one of them up if you happen to see it, or special order the cheese from a vendor.