Short answers, pulled from the story.
The original military orders were the Knights Templar, the Knights Hospitaller, the Order of the Holy Sepulchre, the Order of Saint James, the Order of Calatrava, and the Teutonic Knights. They arose in the Middle Ages in connection with the Crusades, primarily to protect Christian pilgrims and defend the Crusader states.
Pope Clement V formally suppressed the Knights Templar via the papal bull Vox in excelso on the 22nd of March 1312, following trials initiated in France. The order's property was transferred to the Knights Hospitaller in most of Europe, though King Dinis I of Portugal founded the Order of Christ in 1317 for surviving Templars.
The Sovereign Military Order of Malta is the Catholic successor of the Knights Hospitaller. Since 1834 its government has been seated in Rome, where it holds extraterritorial rights. It maintains diplomatic relations with more than one hundred states and the European Union, and holds permanent observer status at the United Nations.
Military orders provided garrisons in Old Livonia from the early thirteenth century and defended the German commercial centre of Riga. The Teutonic Knights absorbed the Livonian Brothers of the Sword and the Order of Dobrzyń between 1229 and 1290, subjugated most of the Baltic tribes, and established a monastic state until their defeat at Tannenberg in 1410.
The Knights Templar was founded around 1118, with Hugues de Payens among its founding figures and Bernard of Clairvaux closely associated with its formation. Pope Honorius II formally recognized the order in 1129.
The Sovereign Military Order of Malta and the Order of Saint John, successors of the Knights Hospitaller, remain active, alongside the Equestrian Order of the Holy Sepulchre of Jerusalem. Those that survive today have evolved into honorific, ceremonial, or charitable organizations rather than military forces.