The famous “left bank” bohemian area, part of the 6th Arrondismant. It grew up around the Abbey of Saint-Germaine-des-Prés, now destroyed except for the Église (church) de Saint-Germain-des-Prés. It was founded by Saint Germaine (c. 496 – 28 May 576), known as the “Father of the Poor”, which sounds a little dodgy to me.
Wander about the back streets. Don’t eat in the Boulevard St. Germaine tourist traps in my opinion, although sitting facing the street at Le Deaux Margots (once frequented by de Beauvoir, Sartre, Hemingway, Camus, Picasso, Joyce, Brecht and such) with a Kir watching the passing parade is lovely. There are lots of cute little restaurants about off the main drag. Rue Mouffetard is nearby, which is pleasant to wander along. Some claim it is the oldest street in Paris.
There are also many and varied museums and churches in the area, and also the lovely Luxembourg Gardens.