It didn't originally stand for peace in general. It was designed by an Englishman named Gerald Holtom. He created it as a symbol for an anti-nuclear arms group called Direct Action Committee Against Nuclear War. So it was originally a "no nukes" symbol (note the rocket-like figure within the circle). The symbol's first public appearance was in a protest march from Trafalgar Square to Aldermaston (where nuclear research was being done).
Americans affiliated with the British anti-nuclear movement brought the symbol to the United States in the late 1950's. When peace activists turned their attention to the Vietnam War in the sixties, they kept the symbol and it's meaning morphed into the general concept of "peace". It retains that meaning today.
Holtom's innovative symbol was a masterstroke. It is at once both a striking symbol and a simple one. Anyone can draw it in just seconds, and everyone instantly recognizes it and knows what it means. Modern corporations would pay a ton of money for a symbol so simple, so powerful and so instantly recognizable.
HAPPY BIRTHDAY to the Peace Symbol! I hope it has a long life, and it's meaning takes hold all over the world.