A central bank is the entity responsible for the monetary policy of a country or of a group of member states. In addition to distributing banknotes and coinage, it is primarily responsible for maintaining the stability of the currency and managing interest rates.
In the case of the French region, the various central banks are of particular interest because they often were issuing notes in common designs for a geographic region which is different from the countries we now have on our maps (or that are presented in paper money catalogs). Viewing the notes sorted by central bank rather than country is often more enlightening, and clears up the source of the common designs (for African notes, especially). I have provided below a list of the relevant central banks, each with a link to the banknotes produced by that bank.
Banque de France
The BdF is the central bank of France. It was formed in 1800 and issues banknotes to this day. BdF has provided financial oversight (often called the "bank of banks") for all of France's territories through the years, coordinating with the central banks listed below, and stabilizing exchange rates from French Francs to CFA, CFP and now Euros. The Banque de France is one of the few central banks that actually prints its own banknotes, with facilities in Chamalières, France.
The Banque de l'Indochine was founded in 1875, initially with branches in Paris, Saigon, French India and Cochinchina. The bank expanded over the years, to include Tonkin, Haiphong, Phnom Penh and Cambodia, Tahiti and Singapore. A branch was even opened in Djibouti. The Banque de l'Indochine was eventually closed in 1952. Virtually all banknotes were provided by Banque de France, denominated in Francs or Piastres.
From 1941 to 1958, the Caisse Centrale was responsible for issuing banknotes in CFA to French West Africa, French Equatorial Africa, Cameroun, Madagascar, the French Overseas Departments (DOM) and Saint Pierre-et-Miquelon. In 1959, the banking functions were transferred to the Institut d'Émission des Départements d'Outre-Mer and the Caisse Centrale has transitioned to be an economic development organization for the regions.
The IEDOM is the French Central Bank currently servicing the French Overseas Departments (DOM) of Guadeloupe, Guiana, Martinique, Mayotte, Reunion and Saint-Pierre-et-Miquelon. The IEDOM was created in 1959, and replaced the Caisse Centrale de la France d'Outre-Mer. From 1959 to 2002, it issued common design banknotes in French or CFA Francs, and as of 2002 is circulating Euros among the DOM.
The IEOM is a French Central Bank created in 1967 and chartered to issue currency for the French Territories in the Pacific (TOM) including New Caledonia, French Polynesia (Tahiti) and Wallis-et-Futuna. Common-design banknotes in CFP Francs are used to this day; New Caledonia's notes are overprinted "NOUMEA", Tahiti's notes are overprinted "PAPEETE".
The BCEAO was created in 1962 to provide monetary policy and delivers banknotes in CFA for countries in the West African States of Benin, Mali, Senegal, Burkina Faso, Ivory Coast, Togo, Niger, and Guinea-Bissau. It currently issues banknotes in CFA, interchangeable with the Euro.
Founded in 1903 in Dakar, Senegal to succeed the Banque du Senegal, the BAO was the central bank for the French West Africa. The BAO continued its to function until 1955, when it transitioned responsibility to the Institut d'Emission de l'A.O.F. et du Togo. In 1959, this function was then succeeded by the Banque Centrale des États de l'Afrique de l'Ouest.
The BEAC (Bank of the Central African States) was created on November 22, 1972 to provide monetary policy including a common banknote series for the countries of the Union Douanière et Économique de l’Afrique Centrale (UDEAC) which includes Cameroon, Central African Republic, Chad, Congo, Gabon, and Equatorial Guinea. In 1999 the UDEAC was superseded by the Communauté Économique et Monétaire de l'Afrique Centrale (CEMAC). The BEAC still functions today, still issuing notes in CFA (convertible to Euro).