After Poland fell in 1939, Europe was at war. But something strange was going on: no one was fighting. Soon. Not yet. The US (neutral) was selling weapons and supplies to those who were becoming The Allies, and there were a couple minor naval skirmishes, but no one was doing anything beyond that. Oh, and the Soviets attacked Finland. No one else got heavily involved.
Winston Churchill called this strange calm the 'Twilight War'. the Germans called it the 'Sitzkrieg' (a sitting war), punning on the Blitzkrieg. the French referred to it as 'drôle de guerre' (a funny/strange war) and the Polish used 'dziwna wojna (a strange war). The Brits favoured the witticism 'Bore War' (as opposed to the Boer War).
Apparently we were working on our humour skills rather than on fighting. Not that they're mutually exclusive.
Anyway, this quiet little war is generally said to have started either on September 3, 1939 when France, United Kingdom, New Zealand, and Australia declared war on Germany or on October 6, 1939 when Poland's last army was defeated. Oh, and Poland likes to focus on September 12, 1939 when the Anglo-French Supreme War Council decided 'Screw Poland. Let's hide from the Germans.' (Well, something along those lines.)
It ended on May 10, 1940 when Nazi Germany invaded Belgium, The Netherlands, and Luxembourg. Either that or on April 9, 1940 when Nazi Germany invaded Denmark and Norway. The beginning and end aren't well-defined.