The word comes from the Latin ministrare—to serve. Originally it was whatever you got to eat, which for most people meant some combination of vegetables and pasta or bread. Later, it came to mean a first course, in broth or not; on restaurant menus you'll very often see minestre as heading for the first courses instead of primi piatti. These might be divided into minestre asciutte (dry, including pasta dishes and risottos) and minestre in brodo, soups.
But nowadays most Italians generally understand minestra as soup, usually a light soup with vegetables and no meat. The Italian-American minestrone just means a big minestra, probably not followed by a second course (in the way a meal of just spaghetti is called a 'spaghettata'). A minestrina is just a simple broth.
The region most devoted to soup is probably Piemonte. A minestra alla Piemontese will probably have lots of beans, cabbage, pork and vegetables. Other noteworthy local soups are based on ceci (chick peas), fontina cheese, tripe, beans, cardoons, walnuts etc.
Image by Alpha