1) Before WWII he and another man, whose name I forget, showed that if the fusion stopped in a sufficiently massive star it would shrink under its own gravity and no physics known then could halt it. It would shrink right down to a mathematical singularity and become a black hole. This paper was ignored because influential physicists of the time didn't believe a singularity was possible.
2) During WWII he was made head of the scientific part of the project to build an atomic bomb, code named the Mnhattan Project. It was he who picked the site for it, up in the mountains at Los Alamos New Mexico. When he viewed the test bomb explosion in July 1945 he thought of a line from the Baghavad-Gita, "I am become death, the destroyer of worlds." Later he said that with nuclear weapons, Physics had known sin.
3) After WWII a controversy arose over the possibility of an H-bomb. Teller, who had been at Los Alamos with Oppenheimer passionately believed it could be built. Opeenheimer was a skeptic. When the red hunt of those days showed that some of Oppenheimer's close relatives had been Communists, he was investigated. Teller famously told the investigators he didn't have complete faith in Oppenheimer's loyalty, and Oppenheimer lost his security clearance. He became the President of the Institute for Advanced Study (home in his day of Einstein, Goedel, and John Nash among many others).