The idea's full name was Project Orion, and it involved both organizations, like say the U.S. Air Force (USAF), DARPA, and NASA, and individuals known for their crazy ideas, like the man behind the Dyson sphere megastructure, Freeman Dyson.
Because the atomic and nuclear power craze was just getting started in the 1950s, when Project Orion first came to light, it was this source of energy that was envisioned as the main means of propulsion for the spaceship. Only it was not meant to use some sort of nuclear reactor to go to and through space, but actual nuclear bombs meant to go off behind it and accelerate it.
This approach is called nuclear pulse propulsion, and it effectively involves detonating nuclear devices to generate thrust. It's something that was never tried in the real world, but at least on paper should generate a lot more thrust and specific impulse than conventional means.
There is though one major issue with nuclear pulse propulsion, and that is fallout from atomic or nuclear detonations. This made Project Orion never leave the drawing boards (model tests were however conducted), but that doesn't stop animation specialist Hazegrayart from putting together an animation of the thing in action, just to see what it would have been like.
Back in the days when the idea was being considered, three different Orions were envisioned, namely the Satellite, Midrange, and Super. It's the Super you can have a look at in the animated video.
Just to give you an idea of what you're seeing, think about the fact the Super Orion was supposed to have a diameter of 400 meters (1,321 feet), and a ship mass of eight million tons. For it to be able to gain forward motion and accelerate, it would have needed to use over 1,000 nuclear bombs, exploding against detonation plates.
With this size, the Super Orion would have been more than an interstellar ark than something our species would have used to move in the immediate vicinity of the solar system. Also, this type of Orion could not have been launched from Earth, but would have had to be built and sent on its way from space. The animation video however show it depart a planet (a number of Orions, actually), and the sight is spectacular.