Vichy France is the unofficial name for the "French State" (État français), successor state of the Third French Republic, named after the de facto capital Vichy, a small spa town. Head of state is the former prime minister and hero of the First World War, Marshal Philippe Pétain. Theoretically, his Regime controls the whole area of the State excepting Alsace-Lorraine and some Italian border areas, in Practice the Wehrmacht controls 60% of it, including the whole Atlantic coast. Which means that the Kriegsmarine can use the French harbors for a more effective war against Britain - and the Luftwaffe has come dangerously close to the British cities.
The new Regime took over after the armistice with Nazi Germany on June 19th in 1940. So far, a big part of the French population is standing behind it (even some "intellectuals" like André Gide). Some prominent Figures like ex-prime minister Pierre Laval, Admiral Darlan and the Generals Weygand and Huntziger support the Regime too - some of them more, some of them less so. The reason for their support is that many Frenchmen think that the Brits left them in the rain, since they soon evacuated ten thousands of their soldiers from Dunkirk and other places. The fact that Britain attacked the French Fleet at Oran on June 27th and killed over 1,000 Frenchmen didn't help either, and the diplomatic relations between them were broken because of this. On the same day, the Brits captured most French ships in harbors of the British Empire to prevent the Axis from getting them instead. The captured ships were mostly given to the Free French naval forces. And on July 2nd, Britain attacked Oran and Dakar again.
The new regime is disunited about the future of their state. Pétain would prefer to stay neutral, Laval is for open support of the Nazis. At least, they agree to get as much as possible from collaboration (e.g. for France).