Short answers, pulled from the story.
The War of the Fourth Coalition was a conflict fought from 1806 to 1807 between Napoleon's French Empire and a coalition led by Prussia and Russia, joined by Saxony, Sweden, and Great Britain. France defeated the coalition, and peace came through the Treaties of Tilsit in July 1807.
Prussia declared war on France on the 9th of October 1806, joining a renewed coalition after growing alarmed at French power following Austria's defeat and the creation of the Confederation of the Rhine. Prussia had also learned that Napoleon secretly planned to hand Hanover, a territory it wanted, to Britain in exchange for peace.
At the double Battle of Jena-Auerstedt on the 14th of October 1806, Napoleon defeated a Prussian force at Jena while Marshal Davout's outnumbered corps routed the main Prussian army under the Duke of Brunswick at Auerstedt. The Duke of Brunswick was mortally wounded in the fighting, and the defeat effectively destroyed Prussia's army as a fighting force.
Prussia suffered 65,000 casualties in its campaign against Napoleon, including the deaths of two members of its royal family, and lost a further 150,000 men as prisoners. It also lost over 4,000 artillery pieces and more than 100,000 muskets stockpiled in Berlin, against roughly 15,000 French casualties for the whole campaign.
The Treaties of Tilsit, signed in July 1807, made peace between France and Russia, which agreed to join the Continental System. Prussia lost territory along the lower Rhine west of the Elbe and land from the former Polish-Lithuanian Commonwealth, which became the Kingdom of Westphalia and the Duchy of Warsaw.
Britain and Sweden remained at war with France even after the Fourth Coalition's defeat. Napoleon later moved to invade Portugal to force it into the Continental System, and after signing the Treaty of Fontainebleau on the 27th of October 1807, French forces began what became the Peninsular War.