The simplest definition for an independent restaurant is that the restaurant is owned by an individual, a family, or a group of people, and it is the only one of its kind. The owners make all of the decisions about the menu, about the appearance and location, and the staff. However, I must immediately contradict myself, as I have eaten in restaurants that I would consider to be "independent" knowing that the owners have several other restaurants, by the same name, in other towns.
Perhaps a better definition, then, for an independent restaurant is that the decisions regarding the menu and so forth are determined by individuals rather than faceless corporate executives, as occurs with chain restaurants.
Another definition of an independent restaurant might be that a dining experience in one is not replicated elsewhere.
Although the idea of an independent restaurant is difficult to quantify, I do know one when I am in one, and for that I am thankful.