Short answers, pulled from the story.
The Knights Hospitaller were formally founded on the 15th of February 1113, when Pope Paschal II issued the papal bull Pie postulatio voluntatis to Blessed Gerardo Sasso, head of the Hospital of St John in Jerusalem. The hospital itself had been established in the late 1060s by Benedictine monks connected to Amalfian merchants.
The Great Siege of Malta took place in 1565, when Sultan Suleiman the Magnificent sent roughly 40,000 men and 400 ships to expel the knights from the island. Grand Master Jean Parisot de Valette refused to abandon the defense despite losing about half his knights. The Ottomans withdrew on the 8th of September after their morale collapsed and Sicilian reinforcements arrived; around 15,000 of the original Ottoman force returned to Constantinople.
The Order was based in Jerusalem and Acre until 1291, then in Kolossi Castle in Cyprus from 1302 to 1310, Rhodes from 1310 to 1522, Malta from 1530 to 1798, and Saint Petersburg from 1799 to 1801. The Sovereign Military Order of Malta settled in Rome in 1834, where it remains today.
Napoleon invaded Malta in June 1798. He arrived on the 9th of June and demanded entry to the port; Grand Master Ferdinand von Hompesch zu Bolheim refused to allow more than two foreign ships at a time. Napoleon ordered an invasion, French troops attacked on the 11th of June, and Hompesch surrendered on the 12th of June. Hompesch resigned as Grand Master on the 6th of July 1799.
The Tribute of the Maltese Falcon was the annual payment the Knights Hospitaller agreed to make to Charles V in exchange for the grant of Malta, Gozo, and the port of Tripoli in 1530. The fee consisted of a single Maltese falcon, to be delivered on All Souls' Day to the Viceroy of Sicily, the king's representative.
Five organizations continue the tradition and have mutually recognized each other: the Sovereign Military Order of Malta, the Most Venerable Order of the Hospital of Saint John, the Bailiwick of Brandenburg of the Chivalric Order of Saint John, the Order of Saint John in the Netherlands, and the Order of Saint John in Sweden. The Sovereign Military Order of Malta is the most senior, maintaining diplomatic relations with 112 countries and operating in 120 countries.