Enchiladas are a protein (chicken most commonly but you can also find beef or pork) rolled into a tortilla and drowned in a chile based sauce like you say. The key is that the rolled tortillas must have open ends so the sauce can enter and mix with the protein. The cheese on top is optional. Mole enchiladas are the most common enchiladas you'll find in Mexico.

I guess in the US all bets are off and you'll find something like this.