It started with a list: "140 Reasons Why Square Will Fail."

In 2009, Jack Dorsey distributed the list, replete with counterpoints, to potential investors to prove that he knew exactly what his new start-up was up against, and more important, that he knew how to face it. Square's plan was to use technology to allow anyone, anywhere to take credit card payments. The list was a cheeky reference to Dorsey's last company, Twitter, and a clever nod to the naysayers who doubted that the guy who invented a microblogging service could take on the financial industry. "I didn't have a finance background, my co-founders didn't, and we didn't hire anyone who worked in finance until we reached 25 employees," Dorsey says. "We get to design what we want to see in the world rather than doing what other people think should be done."

The market buys in

One guiding philosophy at Square is that technology should be so intuitive that no one stops to question how difficult it must have been to create. That philosophy is paying off. In 2010, Square introduced the Card Reader, a 1-inch-square white dongle that plugs into a headset jack and, voilà, turns any iPhone, iPad, or Android phone or tablet into a credit card processor. Today, more than three million merchants are using the Card Reader to process $12 billion a year in transactions. The reader, which is free to order on the Square site, connects to Square Register, a souped-up point-of-sale app that replaces the traditional cash register, allowing customers to pay with cash or swipe a credit or debit card and sign on the screen with a finger. In 2011, Square launched a consumer-facing app, Square Wallet, that allows customers to walk into a store and pay by just saying their names. It's being used at more than 250,000 locations in the U.S. and Canada.

"The idea for Square came from a frustration, one that many small-business owners have struggled with." --James Dyson

Square charges merchants just 2.75 percent per swipe or, for merchants who swipe less than $250,000 a year, a flat fee of $275 per month. For small-business owners accustomed to hidden fees and rates that can run in the ballpark of 4 percent per swipe, this transparency is the cherry on top.

Despite Square's rapid growth, which recently earned the company a valuation of $3.25 billion, there's not a single salesperson among its 500 employees. "The product sells itself," Dorsey says. That fact was apparent at the Artists & Fleas pop-up craft fair in downtown Manhattan, which Dorsey visited one Friday afternoon in March. "It's fantastic!" a thoroughly tattooed, classic New York character named Tony Stinkmetal gushed. He and his friend Keith ("Golly") Bishop run Golly NYC, a company that sells custom superhero T-shirts, housewares, and clothes made of children's sheets and sleeping bags. Stinkmetal says he started using Square because his last credit card processor took several days to deposit money into his account. Square does it overnight. "Everyone who has a business right now is kind of scrambling," he says, "so those two days make a big difference." Mike Lindwasser, a photographer with a booth at the market, praised Square's analytics. Standing nearby as Lindwasser tells his story, Dorsey looks proud. This is why he started Square.

Creation myth

As the oft-told story goes, the idea for Square came about after Jim McKelvey, Dorsey's friend and former boss, who is a computer engineer and skilled glass blower, missed out on a $2,000 sale of one of his glass faucets. McKelvey, like millions of small-business owners in the country, couldn't accept his potential customer's credit card. So he, Dorsey, and an iOS engineer named Tristan O'Tierney developed what would eventually become Square to ensure this would never happen again. "I knew what it was like to be a small merchant and be excluded from commerce because I didn't have a credit card system," says McKelvey, who now sits on Square's board.

As it turned out, Dorsey, McKelvey, and O'Tierney's dream directly violated the major credit card companies' rules. At the time, they banned so-called payment aggregators, which allow people to accept credit cards without a merchant account. (The aggregator acts as the merchant.) The card companies had made an exception for PayPal, an online aggregator, but mobile aggregators were a different story. If Square was going to work, the card companies would have to rewrite their rule books. So Dorsey took a clunky but functional prototype to the major banks and credit card companies and gave them a demo. Within six months, they came around.

In many ways, Square is building the quintessential technology of a new generation. If the past decade was about the iPad, the iPhone, and other gadgets we hold in our hands, Square's success suggests the next decade will be about the technology we never see. Instead, we will merely accept, and come to expect, that it's working all around us. That's the future Dorsey imagines.

The product that edges closest to that future is Square Wallet. Dorsey challenged his engineers to build a product that would let him walk into a coffee shop, place an order, tell the barista his name, and walk away wondering if he even paid. Square Wallet does just that. Users open the app and choose a participating merchant. When they arrive, the merchant has a photo of them on Square Register. With the tap of a button, the merchant can charge the customer's account.

The competition wakes up

Square is the golden child of Silicon Valley. So far, it has raised more than $341 million in financing from VC firms, Starbucks, and other investors. But its success has also bred fierce competition. There are now thousands of players in the mobile payments market, according to Gartner analyst Sandy Shen, including the very credit card companies that once blocked aggregated transactions. Bank of America recently undercut Square's prices with its mobile card reader, which charges vendors 2.7 percent per swipe. And Visa signed a deal with Samsung to equip Samsung's next generation of phones with Visa's PayWave app. PayPal, Intuit, and Groupon are also on Square's ever-growing list of competitors.

For now, Square clearly has the first-mover advantage, especially with small businesses. And last year, it struck a deal with Starbucks to roll out Square Wallet in 7,000 Starbucks stores, which is raising awareness among mainstream consumers. "We're trying to keep our stress levels low and just do innovative things," says Dorsey.

Within the next 10 years, Dorsey expects Square to be available worldwide. That's a tall order, but few people could have predicted the huge success of Twitter when Dorsey co-founded that company in 2006. Two years later, he was publicly ousted as CEO. A year after that, he founded Square. "You fail and you get back up," he says. "That's part of doing anything that people find to be revolutionary or important. You'll have tons of people saying, 'No, you can't do this,' but you do it again, anyway, and you don't let them stop you."