Short answers, pulled from the story.
The country was officially renamed the Hungarian People's Republic on the 20th of August 1949. This change occurred when the newly elected National Assembly passed a constitution nearly identical to the Soviet model.
János Kádár replaced Imre Nagy as head of the newly formed Hungarian Socialist Workers' Party following the Soviet intervention in November 1956. Imre Nagy remained imprisoned until being executed in 1958 alongside other ministers such as Pál Maléter and Géza Losonczy.
In 1966, the Central Committee approved the New Economic Mechanism moving away from strictly planned economy toward decentralized Yugoslav-style system. The government also decriminalized homosexuality in 1961 and achieved lasting economic reforms by the early 1980s while pursuing foreign policies encouraging trade with the West.
An estimated 20,000 people died during the uprising, nearly all during Soviet intervention on the 4th of November 1956. Fighting occurred throughout the country but Hungarian forces were quickly defeated by Red Army tanks that captured airfields and highway junctions within hours.
The Communist Party abandoned power just over a year after János Kádár's retirement in 1988 paving way for free elections in 1990. A four-man executive presidency replaced the Politburo in June 1989 and Parliament adopted constitutional amendments between October 16 and 20 1989 to transform the state into a multiparty democracy.