Tomato sauce
Tomato sauce is a condiment made with tomatoes, and sometimes also meat, onions, basil, salt, oil, garlic and various spices. A few tomatoes are broiled, skinned and mixed with a small amount of chopped meat, onion and some salt, oil, basil and other spices. This mixture is boiled and is often passed through a sieve before it is packaged or used.
The above description details one of literally hundreds of different ways a tomato sauce can be produced.
Marinara sauce (from Italian alla marinara 'sailor style') is another term for a simple tomato sauce for pasta made without meat and usually including tomatoes, onions and herbs. This strongly seasoned sauce is spicy, but not to the degree of fra diavolo. It can be used for any dish that requires tomato sauce, and is generally quicker to prepare than other tomato sauces. It is often used as a dipping sauce for foods such as calzone and fried mozzarella sticks.
In Australia, New Zealand, India and Great Britain the term "tomato sauce" normally refers to the condiment otherwise known as tomato ketchup. In these countries, other sauces made with tomatoes are more usually referred to as pasta sauce etc., depending on their uses. In Australia, the term marinara often refers to a tomato-based seafood dish such as spaghetti marinara or marinara pizza.
Some Italian Americans use the term "gravy" to refer to tomato sauce. "Sunday gravy" is a common type of long-simmered tomato sauce containing meat (often pork or meatballs; similar to an Italian ragù) that is often identified with Italian-American home cooking. It is generally served over pasta. Others just use the term "sauce" to refer to tomato sauce.
Pizza sauce generally refers to a thick, smooth sauce used as a pizza topping. It is similar to but not identical to a marinara sauce, and they are not considered interchangeable.
Escoffier included a tomato sauce recipe using salt pork, butter, and a liaison of wheat flour as one of the mother sauces in his master work, Le Guide Culinaire, but it is not commonly used in French cuisine.
Source: The Cook's Decameron: A Study In Taste, Containing Over Two Hundred Recipes For Italian Dishes from Project Gutenberg. This is from a very old source, and reflects the cooking at the turn of the twentieth century.