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.
minestra di castagne e riso: a thick soup of rice, milk and chestnuts, from the Valle d'Aosta.
minestra di farro: spelt soup
minestra maritata: 'married' Neapolitan soup. Meat (pork and bacon) is 'married' to the vegetables
minestra alla milanese: a thick soup of vegetables, rice, meat and white beans
minestra di uovo: Chinese egg drop soup
minestra di verdure: vegetable soup
minestra di zucca alla milanese: very thick purée of squash with milk and pasta
Images by: Alpha