Much like France itself, the colonies initially were split among support for the Vichy regime and for the Free French forces under de Gaulle.

When the Prime Minister of France - Paul Reynaud - resigned from his position in 1940, Philippe Pétain was placed in charge and he went on to lead the Vichy regime in negotiating peace terms with Germany. When 27 parliamentarians fled to Algeria after refusing to vote in a motion to dissolve the Third Republic, the government in Algiers followed directions from Vichy in confining the escapees to their ship for several days and then releasing them under surveillance some time later. Collaboration with Nazi Germany and Vichy France by the government in North Africa ended with Operation Torch, wherein British and American soldiers invaded key ports in Morocco and Algeria. In this operation the Allies managed to capture François Darlan, leader of all the French forces in North Africa. He successfully ordered French resistance to the Allies in North Africa to cease in exchange for being made High Commissioner.

French colonies in West Africa initially joined the Vichy regime as well - despite efforts by British, Australian, and Free French forces to overthrow the pro Vichy governments in 1940 in the Battle of Dakar. After Operation Torch, these colonies in West Africa joined the Allies.

Colonies in Equatorial Africa (now making up the countries of Chad, Central Africa Republic, Gabon, and Republic of the Congo) were more split - some holdouts proclaimed allegiance for the Vichy regime, though they were short lived. In October and November of 1940, Charles de Gaulle led Free French forces throughout the region and defeated numerous Vichy garrisons before bringing an end to the last pro Vichy holdout in Port Gentil on the coast of Gabon. Some people like Félix Éboué, a colonial administrator in Chad, were defiant in their support for the Free French forces, and Éboué in particular helped ensure much of Chad remained loyal to the Free French when France surrendered.

In the early summer of 1941, Syria fell to an eclectic selection of British, Indian, Australian, Free French, and even Czech soldiers who had invaded in an effort to overthrow the Vichy regime to prevent it from supporting Italian advances into Egypt.

Some French colonies remained loyal to the Free French forces throughout the war, for example the French holdings in India. Louis Bonvin, governor general of French India during WWII immediately proclaimed support for the Allies when France surrendered to Germany. French Polynesia also supported the Allies throughout the war.

By the end of the war, all former French colonies - except for French Indochina which remained under the control of the Japanese - had by one way of another joined the Free French cause. Ironically, some these countries which had so determinedly fought for one French government or another began a series of rebellions and independence wars shortly after the end of the Second World War, such as those in Algeria and Madagascar.

Most of the information came from notes taken in my AP World History class, though I fact checked some of it at this website: http://worldatwar.net/timeline/france/empire40-45.html