Vichy France was the name given to the Nazi puppet government that ruled France between July, 1940 and August, 1944. Philippe
Putain Pétain, a former general in the French Army and the hero of the World War I battle of Verdun, was installed as ruler of the country during this period and actively collaborated with Nazi Germany, including assisting them in carrying out the Holocaust. In theory, Pétain's rule extended across all of France, but in reality, northern France and the Atlantic coast were under military occupation by Nazi Germany, and Pétain's regime ruled over southern France and most of France's colonies. The reason the Nazis allowed a French rump state to exist rather than occupying the country completely was to prevent France's vast colonial empire from falling into Allied hands; regardless, most colonies would either rally to the Free French or, in the case of Indochina, fall under Japanese occupation during the war.
The name Vichy France is used because the seat of French government during that time was temporarily moved from Paris to the resort and spa town of Vichy, and also to distinguish Pétain's collaborationist regime from the French government in exile led by Charles de Gaulle. When writings about World War II refer to "France" or "French troops" those usually refer to the Free French resistance led by de Gaulle, and "Vichy France" is used to refer to Pétain's regime.
The Nazis put Vichy France under military occupation in 1942 (Operation Anton) after the Anglo-American allies occupied Vichy-controlled Algeria and Morocco, though the Vichy regime was allowed to maintain control over France's civil administration. D-Day (June 6, 1944) was the beginning of the end of the Vichy regime, during which troops from the United Kingdom, the United States, Canada, and de Gaulle's Free French troops jointly invaded France at Normandy to drive the Nazis and Vichy regime out. A subsequent invasion along the coast of Southern France (Operation Dragoon) enveloped the remaining German forces in France and liberated the remainder of France.
Philippe Pétain was tried after World War II and sentenced to death, which was later commuted to life in prison in recognition of his previous patriotic work and his age (Pétain was 89 years old at the end of the war).
"Vichy" today is used to mean collaborationism of any sort—at the risk of Godwin's law, of course ("Vichy Democrats", for example, for Democrats who supported George W. Bush's drive to invade Iraq.) Vichy itself is apparently still quite a nice place for a quiet holiday.