Here are The Professor's picks for the best neighborhoods in Paris:

Latin Quarter
Métro:  St-Michel,  Cardinal Lemoine, Cluny-La Sorbonne, Place Monge and more.
The area on the left bank that has traditionally been the home of students, classic cafés, and a lively village-like atmosphere.  It is a great walking neighborhood.  I would compare it to the East Village in New York City.  This is my most favorite place to visit and stay.

Île Saint-Louis
Métro:  No métro station on Île Saint-Louis.  Cité is closest.
This is a traditional Parisian neighborhood of cobblestone streets, cafés, bistros, pâtisseries, boulangeries and quaint shops.  It is just over the bridge from Notre Dame and Île de la Cité.  Some charming small hotels and B&B’s are located here.  Get a true flavor of Paris while walking through this neighborhood.  Cross the bridge (often there are street performers here) from Notre Dame to get there. Have ice cream at Berthillon’s.  Afterwards, take a stroll down the Seine towards the Eiffel Tower.

Marais
Métro:  St-Paul.
This is the Jewish and the gay section of Paris.  It is a revitalized neighborhood that has regained prominence in the arts and culture and in dining.  It is a charming walking neighborhood.  Eat a falafel at L’as du Fallafel on rue des Rosiers and have a pain au chocolate from one of the bakeries.  The Picasso Museum, Place des Vosges, and Carnavalet (history of Paris museum) are in the neighborhood.  3 hours.  Enjoy a full day if planning to visit the museums.

Saint Germain des-Prés
Métro:  St Germain des-Prés.
Adjacent to the Latin Quarter on the left bank.  It is the home of chic shops, time-honored restaurants, shopping, and trendy art galleries.  There are many good, moderately priced hotels in this area.  It is a sophisticated area retaining many of the characteristics of a traditional French neighborhood.  If looking for an area to comfortably walk to a mix of restaurants, shops, cultural locales, markets, gardens, and theaters – Saint Germain des-Prés is for you.

For more information on Paris tours, see www.travelingprofessor.com