Provencal in Provence*
Langue d'Oc in Languedoc*
Basque in the Basque country*
Breton in Brittany*
Alsatian in Alsace*
Flemish in French Flanders*
Normand in Normandy
Burgundian in Burgundy
Gascon in Gascony*
Artésien in Artois*
I have starred those that still have local dialects, but French is now pretty well uniform all over France. The Basque language is a completely different language from French altogether. Alsatian is markedly Germanic. Breton is Gaelic. Flemish is akin to Dutch. Provencal and Langue d' Oc are ancient French Southern dialects. Artésien has had a strong input from Flemish and English because it is the nearest to both geographically, but is hardly spoken nowadays except for some local expressions.
This is only in Metropolitan France. There are French departments overseas that have different cultures altogether and where the local dialects are very different to classic French (Guadeloupe, Martinique, Guyane, La Réunion) and where Créole is spoken by the indigenous population.