Explaining symbols without giving away the story is no small task. I've aimed for a balanced approach, so I apologize if I include anything you haven't yet figured out for yourself. I also apologize if my explanation is a bit simplistic, as I left out some details to avoid spoilers.
In Greek mythology, Atlas was a titan who was sentenced to forever hold up the weight of the entire world. In "Atlas Shrugged," the idea is advanced that certain individuals are given the task of keeping society working and advancing. The members of this class--which includes everything from businessmen to scientists to artists--are distinguished primarily by their moral capacity.
In the novel, these "Atlas-es" start disappearing one-by-one. They can't be found, their possessions remain at home, their bank accounts are untouched, and none of their business or legal affairs are taken care of. They vanish with nothing more than the clothes on their backs. And without them holding up the other structures of civilization, society starts disintegrating. As you go through the book, you see this more and more. The act of refusing to tend to the activities necessary to promote the well-being of human society is symbolized by saying that Atlas is shrugging.
John Galt is hinted at throughout the book. He is the symbol of the disappearances. The fantastic stories about him are all symbolic of a person who gives up everything, although it isn't always clear about what his motivation is. When the real John Galt finally appears, the motivation is revealed. Symbolically, you soon see that he's the fulfillment of all of the things that the protagonists want and admire, while at the same time the embodiment of all that the antagonists fear and hate.