The social movement that is most likely to change the world is the environmental movement.
If this movement succeeds, it could fundamentally change the future of the world. Let us assume that global warming will be a major problem if we do not do something about it. If this is the case, then the environmental movement could really change the world. It could prevent the polar ice caps from melting. It could prevent sea levels from rising and drowning cities. This would be a tremendous impact greater than that of any other movement around today.
I would say the continuing protests from the Arab spring. Major dictatos were ousted from power, and most likely, Syria is next. This is more than just cruel dictators, it is the same area where most of the world's oil supply comes from.