Guernica, of course, is a very famous painting by Pablo Picasso depicting the bombing of Guernica, Spain by the Italians and Germans during the Spanish Civil War. It is famous for exploring the tragedy of war, especially the effect on innocent civilians.
In regards to why the painting was "moved" to New York City, ... the actual story is that the painting was on tour since its creation. It went all over the world, but it traveled so very much that people started getting worried about its state. That was the reason for the decison to keep it in one place: the Museum of Modern Art in New York City. I guess the more specific reason as to why that particular spot on the tour was chosen rested on the fact that many of Picasso's preliminary studies were already housed there.
As for its return to Spain, Picasso absolutely refused to allow the painting back into Spain until the Spanish people enjoyed a republic. He added other lesser conditions a bit later like "the restoration of public liberties and democratic institutions." It was only after Franco's death that Spain became a democratic constitutional monarchy and the painting was returned to Spain, albeit a few years after Picasso's death.