Short answers, pulled from the story.
The Italian Socialist Party was founded on the 14th of August 1892 in Genoa, initially under the name Party of Italian Workers. It adopted the name Italian Socialist Party at its third congress on the 13th of January 1895 in Parma.
The PSI's best ever result was 32.0% of the vote and 156 seats in the Chamber of Deputies in the 1919 Italian general election, making it Italy's first party at the time. The party was led in that election by Nicola Bombacci.
Bettino Craxi was the PSI's party secretary from 1976 and became Italy's first Socialist prime minister in 1983. His government lasted three and a half years, during which it boosted the country's GNP, controlled inflation, and confronted the United States directly in the Sigonella incident. He resigned as party secretary in February 1993 amid the Tangentopoli corruption scandals.
Tangentopoli refers to a wave of judicial investigations called mani pulite, or "clean hands", that began in Milan in February 1992 when a Socialist hospital administrator named Mario Chiesa was caught taking a bribe. The investigations, led by magistrates including Antonio Di Pietro, exposed systemic corruption and illegal party funding across the Italian political system, ultimately bringing down the PSI and several other parties.
Mussolini was expelled from the PSI during World War I because he showed sympathy to national syndicalism and supported Italian entry into the war, a position the dominant internationalist wing of the party opposed. The PSI's refusal to support what it called a bourgeois war led to his departure and expulsion. He later founded the National Fascist Party, which emerged from the Fasces of Combat at the Third Fascist Congress in late 1921.
The PSI was disbanded on the 13th of November 1994, following two years of Tangentopoli scandals. Several successor parties followed, including the Italian Socialists (1994-1998) and the Italian Democratic Socialists (1998-2007). A new Socialist Party formed in 2007 renamed itself the Italian Socialist Party in 2011 and remains the only Italian parliamentary party to explicitly identify as Socialist.