Short answers, pulled from the story.
The Treaties of Tilsit were two peace agreements signed in July 1807 following Napoleon's victory at Friedland and the end of the War of the Fourth Coalition. The first was signed on the 7th of July between France and Russia; the second on the 9th of July between France and Prussia.
Napoleon signed both treaties on behalf of France. Russian Emperor Alexander I signed the Franco-Russian treaty on the 7th of July, while King Frederick William III of Prussia signed the second treaty on the 9th of July.
Prussia lost roughly half its territory, shrinking from 5,700 square miles to 2,800 square miles, and its population fell from 9.75 million to about 4.5 million. Lost areas included Cottbus, which went to Saxony, Białystok, which went to Russia, and most Polish lands acquired through the 1772-1795 partitions, which became the Duchy of Warsaw.
The meeting on a raft moored in the Neman river was chosen to express the perfect equality of both emperors, as the river belonged to neither empire. The city of Tilsit itself was declared neutral for the occasion.
The Continental System was Napoleon's economic campaign against the British Empire. At Tilsit, Russia agreed to join it as a key term of the Franco-Russian treaty signed on the 7th of July 1807, though the alliance broke down by 1810 when Alexander I began allowing neutral ships into Russian ports.
The Franco-Russian alliance formed at Tilsit deteriorated after Alexander I began allowing neutral ships to dock in Russian ports in 1810, violating his Continental System commitment. Napoleon responded by crossing the Neman into Russia in June 1812, ending any remaining vestige of the alliance.