It is a name that refers to a collection of French Resistance Movements. As /u/AzertyKeys pointed out there were various phases of resistance in France but there was no organization called "Resistance" but rather a collection of various organizations summed up under that moniker.

Among them were for example the Gaullist Armée secrète, the communist Francs-tireurs et partisans, and the giraundist Organisation de résistance de l'armée.

As has been previously pointed out, these united under one command in mid-1943 under the Conseil National de la Résistance (CNR) and Jean Moulin, so when the topic comes up in popular media, it is very likely that this is meant by "the Resistance". The CNR consisted of various representatives of the different factions within the resistance and it was only in 1945 that they adopted one common political program after months of negotiation.

Sources:

• Robert Paxton: Vichy France: Old Guard and New Order, 1940-1944 (1972).

• Cobb, Matthew (2009). The Resistance: The French Fight against the Nazis.

• Knight, Frida (1975). The French Resistance, 1940–44.

I'll assume you want an answer specificaly focused on the FFI (forces françaises de l'intérieur ).
There are roughly three parts in the history of the french interior resistance :

• the first from the Fall of France to Operation Barbarossa was mainly made up of nationalists and communists and was extremely disorganized. There are multiple reasons for this but mainly it is due to the fact that there were infighting between both movements as well as within each, for the nationalists (made up mainly of Action Française royalists) it was due to the immense respect that Pétain wielded as well as the fact that most of these people hated the old third republic ("better Hitler than Blum!" Was a common motto). For the communist the problem was that Stalin had strictly prohibited partisan actions from the PCF. In this first part resistance was small, disorganized and strictly limited to the maquis.
• The second part begins with Operation Barbarossa from the. On the PCF puts its immense power base into the resistance such allowed for many more acts of sabotage and more effecie cy a cross a wider region but it still isn't the resistance you know from the movies.
  -the third phase began when Jean Moullin was parachuted from Britain and managed to organize the Conseil National de la Résistance which unified all resistance movements in France and answered to Charles de Gaulle .

This last one is the one you know about from the movies it was extremely well organized for a movement of franc-tireurs and partisans and was extremely valuable to the allies for liberating france. By the time liberation came the Resistance was able to liberate Paris without the help of a single foreign soldier and soon after the newly founded french army had over a million men to invade and occupy germany (although the war was by then very much already over)

As the others mention, the early French resistance was hugely complicated, but the situation you describe, local Resistance forces acting as scouts for the Allied forces, was predominantly coordinated through Supreme Headquarters Allied Expeditionary Force (SHEAF) and General Kœnig's Forces Françaises de l'Intérieur (FFI).

The Western Allies' intelligence services (SOE, OSS, BCRA) had hundreds of agents on the ground that had been organizing resistance activities, and by June 1944 (when Kœnig took over the FFI) they had tens of thousands of fighters, organized into smallish bands, that were being armed via huge American supply drops. Their primary role was as guerrillas and saboteurs, to disrupt German communications and logistics so they couldn't respond effectively to the D-Day invasion of Normandy. As the Allies advanced, the leaders of these groups would try to make contact and provide intelligence on German forces in the area (which wasn't always heeded).

The FFI was particularly effective in helping Operation Dragoon, the invasion of southern France. Henri Zeller, the FFI commander of south eastern France, was able to persuade the General Patch to tear up his original timetable. The FFI was essentially light infantry, and had virtual control of the region. They were able to protect the Allies flanks, so the main forces made huge progress, and reached Grenoble 36 hours after landing, which was their +90 day objective.

Sources:

William Casey, The Secret War Agains Hitler (Casey was in the OSS and eventually became the director of the CIA.)

H.R. Kedward, Resistance in Vichy France

A bunch of original sources. Ask me if you'd like more info.