Short answers, pulled from the story.
The French invasion of Russia, also known as the Russian campaign or the Patriotic War of 1812, was a military campaign initiated by Napoleon to force the Russian Empire back into the Continental blockade of the United Kingdom. Beginning on the 24th of June 1812, Napoleon's Grande Armée crossed the Neman River and advanced deep into Russia, eventually reaching and occupying Moscow before a catastrophic retreat destroyed the army.
Napoleon invaded Russia because Tsar Alexander I broke away from the Continental System, the blockade aimed at Britain, on the 31st of December 1810. This removed Napoleon's primary foreign policy tool against the United Kingdom. Napoleon's stated goal was to compel the Russian Empire to rejoin the blockade and to restore French dominance over neighboring states.
Around a million soldiers and civilians died during the campaign. Richard K. Riehn estimated that 685,000 men marched into Russia in 1812, with fewer than 70,000 known survivors. Half of French losses came from disease rather than combat. Russian field army losses were nearly as large, with Russian forces reduced to 40,000 effectives by the time they reached the Neman.
The Battle of Borodino, fought on the 7th of September 1812, was the largest battle of the French invasion of Russia, involving more than 250,000 troops and causing at least 70,000 casualties. It was the bloodiest single day in the Napoleonic Wars. Napoleon captured the main Russian positions but failed to destroy the Russian army, and declined to use the Imperial Guard, losing his best opportunity for a decisive victory.
Cold weather was a major factor in the retreat but not the primary cause of Napoleon's losses. Charles Joseph Minard's famous infographic shows that French losses were highest during the summer and autumn marches, caused mainly by inadequate logistics, starvation, and disease. Winter arrived on the 6th of November 1812 with the army still wearing summer clothing and with horses unequipped for icy roads, but by that point the campaign had already been lost.
Napoleon entered Moscow on the 14th of September 1812 to find it deserted and set ablaze by military governor Fyodor Rostopchin. Within forty-eight hours, three quarters of the city had burned. Napoleon remained in Moscow for five weeks waiting for a peace proposal from Tsar Alexander that never came. He finally ordered the retreat on the 19th of October 1812.