Short answers, pulled from the story.
The Papal States began in 754 when Pepin the Short led a Frankish army into Italy to defeat the Lombards. Pope Stephen II received lands formerly held by Arian Christian Lombards as a gift from the Frankish king, marking the beginning of papal temporal power.
The Theophylacti family controlled the Papal States during the early 10th century. This era later received the nickname dark age or rule by harlots due to the influence of local Roman nobility over the papacy.
By 1649 after annexing the Duchy of Castro the Papal States reached their greatest extent. They included most of central Italy plus small enclaves like Benevento and Pontecorvo.
By 1792 the Papal army had a mere 5,000 regular soldiers for a state controlling most of Central Italy. Control over the military was divided among multiple commanders who jealously guarded their powers while the navy consisted of only three galleys and two coastal corvettes.
The opportunity for the Kingdom of Italy to eliminate the Papal States came in 1870 when the city was captured on the 20th of September 1870. Rome and what remained of the Papal States were annexed to the Kingdom of Italy following a plebiscite in October marking the definitive end of the Papal States.
The Lateran Treaty with Italy was signed on the 11th of February 1929 between Pope Pius XI and Benito Mussolini head of the Italian government. The treaty created the State of Vatican City as a sovereign territory within Rome ending the Prisoner in the Vatican period that had lasted since 1870.