I'll try to explain simply and briefly :
After the fall of France on 14th June 1940, the German Army occupied Paris. Paul Reynaud, the French prime minister, realized that the German Western Offensive could not be halted and suggested that the government should move to territories it owned in North Africa. This was opposed by his vice-premier, Henri-Philippe Petain, and the supreme commander of the armed forces, General Maxime Weygand. They insisted that the government should remain in France and seek an armistice.
Reynaud resigned and President Albert Lebrun, appointed Petain as France's new premier he immediately began negotiations with Germany and on 22nd June signed an armistice with Germany.
The terms of the agreement divided France into occupied and unoccupied zones, with a rigid demarcation line between the two parts. The Germans would directly control three-fifths of the country, an area that included northern and western France and the entire Atlantic coast. The remaining section of the country would be administered by the French government at Vichy under Marshal Petain.
The French Army was disbanded except for a force of 100,000 men to maintain domestic order. The 1.5 million French soldiers captured by the Germans were to remain prisoners of war.
The French government also agreed to stop members of its armed forces from leaving the country and instructed its citizens not to fight against the Germans, France also had to pay the occupation costs of the German troops.
Over the next four years Petain led the right-wing government of Vichy France. The famous revolutionary principles of "Liberty, Equality, Fraternity" were replaced by "Work, Family, Fatherland".
Many French troops were evacuated to Britain and formed the Free French Army under General De Gaulle, these troops fohgt alongside the Allies in many campaigns.
Much of the French Fleet was berthed at Oran in Algeria and were attacked by the Royal Navy lest they be used by Axis forces in the Mediterranean.
The Germans also allowed France to keep its territories in Syria and Algeria
Syrias position on the Eastern Mediterranean coast made it strategically important for both Britain and Nazi Germany. The Allies also feared that Petain would allow the Luftwaffe to establish air bases in the country.
On 8th June 1941 the British Army and Free French forces entered Syria from Iraq and Palestine. After facing tough resistance from the Vichy forces the Allies captured Damascus on 17th June. The armistice was signed on 12th July and pro-British regimes were maintained in Syria for the rest of the war.
Vichy French Forces also initially opposed the Allies after the Operation Torch landings in North Africa, but were eventually persuaded to join forces with the Allies and opppose the Axis forces in Tunisia
Free French Forces fought throughout the North African campaign alongside the 8th Army and fought some memorable battles particularly at Bir Hacheim.
There were also of course forces opposed to the German occupation that remained under great secrecy in France and organised as the Maquis or Resistance, these fought a long war of covert operations including sabotage and espionage for the Allied cause.
After the D-day landings took place the Maquis and other resistance groups emerged to help in the liberation of their country.
Marshal Petain and his ministers fled to Germany where they established an exiled government at Sigmaringen.
In 1945 the leaders of the Vichy government were arrested and some were executed for war crimes.
I'm sorry if this is a bit long but its a complex history only part of which can be touched on here, but I hope this gives you an insight into the split of French Forces in WW2.
Happy modelling
Ant