The cheese you are describing is likely Parmesan cheese, which is a hard, aged cheese with a mild to moderately sharp flavor. It has a slightly salty and nutty taste, and can sometimes have a smoky flavor due to the way it is produced. In the United States, it is commonly used grated as a topping for pasta dishes, salads, and soups. However, in Italy, it is more often eaten in slices or chunks as a table cheese or used as an ingredient in various dishes.