There are two components to the French contribution to the Allied war effort: the Free French Forces (Charles de Gaulle's regular army-in-exile) and the French Forces of the Interior (the umbrella organisation for the resistance groups in occupied France.)

In June 1944, the majority of the Free French Forces were deployed on the Italian front. De Gaulle's regular forces' major contribution to the liberation of France began in August, when the First Army was withdrawn from Italy and redeployed to participate in Operation Dragoon, the airborne/amphibious invasion of southern France.

Only one FFF division — Philippe Leclerc's 2nd Armoured Division — participated in the Normandy campaign, and not on D-Day; they landed at Utah Beach in mid-August and led the liberation of Paris ten days later. But smaller French combat formations were involved in the D-Day landings and the early Normandy campaign.

One battalion of French commandos was attached to the British Royal Marines, and landed with the British at Sword Beach on D-Day. Two battalions of French paratroopers were attached to the British Special Air Service: one of them dropped into Brittany on 5 June (Operation Dingson) to coordinate FFI forces and harass the German rearguard; the other wasn't deployed in France until July.

But the really interesting part is that played by the FFI — the Maquis resistance groups in France, coordinated by intelligence officers from the British SOE, the American OSS and the French BCRA (Operation Jedburgh). On 5 June, the BBC broadcast a pre-arranged codeword, alerting the resistance that the invasion of France would begin imminently. Over the following days, the FFI carried out ambush and sabotage attacks on German supply lines and communications infrastructure (particularly the railway network), helping to weaken the structural integrity of the German defence in much the same way as the airborne landings did. As the liberation progressed, the FFI were reintegrated into the wider FFF command structure.

There's plenty of debate about how strategically significant the Maquis were — certainly, these were somewhat chaotic operations, but that's commonly the case with (and contributes to the effectiveness of) irregular warfare. But it's generally accepted that Resistance sabotage and Allied special forces operations helped to isolate German forces in Normandy, and hinder effective reinforcement of the Atlantic defensive lines.

Here's an interesting lecture on the SOE and the French Resistance around D-Day. M. R. D. Foot's SOE in France is a good account of (you guessed it) the SOE and Jedburgh operations in occupied France.