This depends on the country. Re-ask your question or see related questions.
UK: Harold Macmillan.
She became prime minister for the first time in 1960.
In 1960, Australia's Prime Minister was Robert Gordon Menzies.
The Prime Minister in Togo between 27 April 1960 to 12 April 1961 was Sylvanus Olympio.
The Prime Minister of the Republic of the Congo in 1960 was Patrice Emergy Lumumba.
The 1960s began with Progressive Conservative John Diefenbaker as prime minister. Diefenbaker was defeated in the 1963 election and Liberal Lester Pearson became Prime Minister. Pearson retired in 1968 and was succeeded by Liberal Pierre Trudeau.
She was the first woman in the world to become a prime minister, when she became prime minister of Ceylon (now known as Sri Lanka), in 1960.
Sirimavo Bandaranaike. She was elected the Prime Minister of Ceylon in 1960, the country was later renamed Sri Lanka
1960: Siramavo Bandaranaike of Ceylon (now Sri Lanka) becomes the world's first female Prime Minister.
Sir Robert Gordon Menzies.
Ferenc Münnich, from Jan 1958 to Sept 1961
Israeli
Hayato Ikeda was the Prime Minister of Japan in 1960. He served in office from July 19, 1960 to November 9, 1964.