On February 21, 1958, designer Gerald Holton created a symbol for the upcoming Easter march from London to the Atomic Weapons Research Establishment near Aldermaston, England. The march was organized by the Direct Action Committee Against Nuclear War, and the symbol represented nuclear disarmament. It was based on the semaphoric symbols for the letteers "N" and "D."
How the Aldermaston symbol evolved into the peace symbol is not exactly clear. According to Wikipedia, Philip Altbach, a student at the University of Chicago, brought a bag of the Aldermaston buttons back to the United States. I've also heard that civil rights and peace activist Bayard Rustin played a role in popularizing the symbol in the United States. In any case, once it got to America, it lost its specific meaning of nuclear disarmament and became a generalized peace symbol.
The symbol has been denounced as the Satanic symbol of a broken cross, the "footprint of the American Chicken," and countless other things. But by its its very simplicity it has endured. It even showed up in a Star Trek novel, "Strangers from the Sky," by Margaret Wander Bonanno, in which Mr. Spock is sent back in time to mid-21st Century Earth and helps save two Vulcans who have crashed their spaceship in the Pacific. Spock visits his human ancestor, Professor Grayson, who gives him a peace symbol on a chain. He calls Spock "Ben" in honor of peace activist Dr.Benjamin Spock. In the story, the peace symbol is something of a magic talisman.
I'm sure it will still be around in 2058.
Note: After listening to "Strangers from the Sky" on my drive from Elkhart back to Bloomington while hoping the "Check Engine" light on my 1990 Toyota didn't mean I was going to crash-land on Interstate 55 (it didn't), I realized the story was much more complex and interesting than I remembered. I've revised the summation of the story, though that doesn't do it justice.