Short answers, pulled from the story.
Moscow, Third Rome is a theological and political concept formulated in the Grand Duchy of Moscow in the 15th-16th centuries, asserting that Moscow is the legitimate successor to ancient Rome and Constantinople as the center of Eastern Orthodox Christianity. The monk Philotheus gave it its most famous expression: "Two Romes have fallen, the third stands, and there will be no fourth."
When Mehmed II captured Constantinople on the 29th of May 1453, the Eastern Orthodox world lost its basileus, its sacred Christian emperor, and its spiritual center. Moscow, as the strongest surviving Orthodox power, became the natural candidate to fill the void. The Ottoman Empire's own claim to Rome's succession was not widely accepted among Orthodox Christians.
Metropolitan Zosimus made the earliest formal declaration in 1492, calling Ivan III "the new Tsar Constantine of the new city of Constantine, Moscow" in a foreword to his work Presentation of the Paschalion. The monk Philotheus gave the concept its most celebrated expression in the early 16th century, declaring that two Romes had fallen and there would be no fourth.
The concept first appeared in an official state document in 1589, when the charter establishing the Moscow Patriarchate was drafted by the Muscovite chancellery and signed by Ecumenical Patriarch Jeremias II. This followed Jeremias II's visit to Moscow in 1588-1589, which had the stated purpose of collecting funds for Eastern Orthodox communities in the Ottoman Empire.
Peter the Great's reforms in the early 18th century largely marginalized the concept. He abolished the Patriarchate, adopted the secular Latin title of Imperator, and moved the capital from Moscow to St. Petersburg, which he envisioned as a "New Amsterdam" or secular new Rome rather than a religious center.
Under President Vladimir Putin, the Kremlin and the Russian Orthodox Church have invoked the Moscow, Third Rome legacy to assert Russia's exceptionalism as the defender of traditional Christian values. Patriarch Kirill articulated a related concept, the Russian world, in a keynote speech on the 3rd of November 2009, describing it as a "common civilisational space." Critics and scholars argue this doctrine functions as an ideological justification for Russian imperialism and has been cited as a driver of the Russo-Ukrainian War.