I think you're asking about what's usually called the Fall of France or the Battle of France, from the major German invasion starting 10 May 1940 to the German-French armistice of 25 June 1940.

Mentioning previous answers is not intended to discourage follow-on discussion -- follow-on questions, information, and debate is welcome.

There's an FAQ section in the World War II section, "Maginot Line and Fall of France", at https://www.reddit.com/r/AskHistorians/wiki/faq/militaryhistory/wwii/minor#wiki_maginot_line_and_fall_of_france . There are four discussions there.

• "If the French had not built the Maginot Line prior to WWII, would they have resisted the German invasion longer?" at https://www.reddit.com/r/AskHistorians/comments/2zijiq/if_the_french_had_not_built_the_maginot_line/ with an answer by /u/DuxBelisarius . You might find this interesting as an aspect of the question.
• "After World War I, France was the most powerful country in Continental Europe. What specifically was it that allowed Germany to so easily crush the French military/ Frances heavily fortified border?" at https://www.reddit.com/r/AskHistorians/comments/2onrnk/after_world_war_i_france_was_the_most_powerful/ , answers by [deleted] and DuxBelisarius. Not as much Maginot.
• "Why did the French fall in WWII? What factors led to this fall and made it possible?" at https://www.reddit.com/r/AskHistorians/comments/292qj7/why_did_the_french_fall_in_wwii_what_factors_led/ . [deleted] goes into more detail, but you might need to start skimming now when they hit points you've already seen.
• "How did german tank armies manage to traverse through the ardennes forest during Fall Gelb? Something that was deemed impossible by the allies." at https://www.reddit.com/r/AskHistorians/comments/2cvgu8/how_did_german_tank_armies_manage_to_traverse/ , answers mostly by Badgerfest and /u/kieslowskifan . There are a lot of other answers here though (I didn't point directly to kieslowskifan and Badgerfest's replies).

Edit: Actually, down below is a section "French military", which also has applicable discussion. Specifically,

• "How capable was the French Army in WWII?" at https://www.reddit.com/r/AskHistorians/comments/161io9/how_capable_was_the_french_army_in_wwii/ with an answer by /u/Badgerfest
• "France's poor showing in WW2 - How valid are William Shirer's theories of communist + fascist sympathy inside the military?" at https://www.reddit.com/r/AskHistorians/comments/2eu6o5/frances_poor_showing_in_ww2_how_valid_are_william/ , with an answer by /u/kieslowskifan
• "When Hitler invaded a country what happen to that country's military? Did it become part of Hitlers' army or broken up?" at https://www.reddit.com/r/AskHistorians/comments/26o9uy/when_hitler_invaded_a_country_what_happen_to_that/ with answers by nate077.
• "French in world war 2" at https://www.reddit.com/r/AskHistorians/comments/3sc0tv/french_in_world_war_2/ , with more answers by DuxBelisarius.

I'll briefly explain "Vichy France" in a reply.