Albert Einstein and his colleagues Podolsky and Rosen proposed a simple way to rid quantum mechanics of its most disturbing feature--called non-locality--in which an action undertaken here can affect the result of a measurement undertaken there, even if here and there are far apart. John Bell came up with a way to test Einstein's vision of reality, ultimately showing that Einstein's vision was wrong.That text is correct, but if you listen to the video, Greene says that the world was proved to be nonlocal. He says measuring the spin of a particle can have an effect on a distant particle.
Greene sees the big issue as to whether a spin measurement is a random event at the time of the measurement, or it is predetermined in advance.
Briefly, quantum mechanics is somewhat strange because electrons act like waves, and you cannot measure their position and momentum at the same time. Einstein and others had an idea for replacing quantum mechanics with a classical theory of hidden local variables, because that would be more compatible with his determinism prejudices. Bell and subsequent experiment proved that all those classical theories do not work. The world is quantum.
Bell did not show any nonlocality. He only helped show that the classical theories don't work. Almost everyone was convinced of that in 1930 anyway.
Watching this video will just get you confused. There is no action-at-a-distance.
Greene is very good at explaining a lot of physics, but he really goes off the rails when he talks about Bell's theorem, many-worlds, or string theory.