While the piazza may be equivalent to a "public square" in English, it doesn't have to be square, or even rectangular in shape. In Lucca, the Piazza dell'Anfiteatro is an open space in a former ampitheater, and takes on its oval shape.
One of the joys of touring Italy is to spend time doing nothing (far neinte) at a caffe located in an historic piazza, just for the people watching.
Events may be held in larger piazze, as well as weekly or daily markets. Piazza delle erbe indicates a piazza used for a vegetable market (this may be historic, and not the current use of the piazza).
A piazza may be set with tables for a sagra, or festival where food will be served, cooked by locals with a passion for cooking.