Twitter's Jack Dorsey working on a pay-by-iPhone system
Square basically turns your iPhone into a pay station. You swipe your card, have the employee enter the amount to charge, and sign with your finger. From there, the cost of your purchase goes straight into the store's bank account, minus a little bit off the top for Square.
The info on Jack's new business was pieced together by Engadget and Mashable, with some help from Coolhunting (who wrote about the Square payment system before anyone knew Jack was involved).
The only rumor I've seen that doesn't make sense here is Mashable's assertion that Jack's new company is called Self Edge NYC. Self Edge is actually an upscale denim shop that started in San Francisco, and it looks they'll be accepting Square payments. Self Edge is listed and linked on the SquareUp website.