Short answers, pulled from the story.
Hafizullah Amin was an Afghan communist revolutionary who served as head of state of the Democratic Republic of Afghanistan from September to December 1979. He held the positions of Chairman of the Revolutionary Council, Chairman of the Council of Ministers, and General Secretary of the PDPA Central Committee simultaneously, making him Afghanistan's supreme leader during that period.
Amin rose to power by defeating an assassination attempt orchestrated against him on the 14th of September 1979, then ordering the arrest of his rival and mentor Nur Muhammad Taraki. On the 16th of September 1979 he assumed all three of Afghanistan's top government positions at once, with his elevation confirmed unanimously by the PDPA Politburo.
The Soviet leadership, particularly KGB chairman Yuri Andropov, concluded that Amin's mass repression had destroyed the Afghan military and government's ability to manage the insurgency, and feared he was shifting Afghanistan toward a pro-United States foreign policy. A Soviet Politburo assessment called him "a power-hungry leader who is distinguished by brutality and treachery," and Operation Storm-333 was launched to remove and replace him with Babrak Karmal.
Soviet special forces assassinated Amin at the Tajbeg Palace in Kabul on the 27th of December 1979 as part of Operation Storm-333. The Soviets had made earlier attempts including poisoning attempts on the 13th of December and again hours before the assault; Amin survived the final poisoning because carbonation in his Coca-Cola diluted the toxic agent. He was killed during the military assault that followed, though the exact manner of his death has never been confirmed.
The Saur Revolution was the coup of the 27th of April 1978 that overthrew the government of Mohammad Daoud Khan and established the Democratic Republic of Afghanistan. Amin was its main organiser, spending years recruiting disaffected military officers into the PDPA and giving the signal for the revolution to begin at 9 am on that day after learning that Taraki had been arrested.
The men of Amin's family were executed either immediately or shortly after his death on the 27th of December 1979; his brother Abdullah and nephew Asadullah were killed in June 1980. The women of the family, including his daughter who had been wounded during the palace assault, were imprisoned at Pul-e-Charkhi prison and were not released until President Najibullah freed them in early 1992.