What happened to Ngo Dinh Diem and his brother Nhu?

Assassinated.

Ngo Dinh Diem was the mandarin of Nguyen dynasty and later was named as the prime minister of the State of Vietnam by Head of state Bao Dai.

Later on, he deposed the Bao Dai and established the first Republic of Vietnam (RVM) with himself as a president.

In November 1963, after constant religious protests and non-violent resistances, Diệm was assassinated, along with his brother, Ngô Đình Nhu, by Nguyễn Văn Nhung, the aide of the leader of The Army of Republic of Vietnam (ARVN) General Dương Văn Minh during a coup d’état sanctioned by the US.

P.S.: Coup d'etat : an attempt by small group to remove the existing government by violence.