For starters, fans of the disappearing-photo-sending service are mainly teens and other so-called digital natives, a fickle (translation: attention-deficient) audience that jumps from one digital confection to the next, bolting the minute a fave app goes mainstream. Then there are the founders of Snapchat — CEO Evan Spiegel is 23, Bobby Murphy, the chief technical officer, is 25.

They’re the kind of superconfident millennials who drive executives crazy because they don’t know what they don’t know. To wit: These kids turned down a multibillion-dollar acquisition offer from Facebook (FB), even though they don’t have a cent of reported revenue. And just what the heck is a “disappearing-photo-sending service” anyway?

But it would be a mistake to dismiss the Snapchat juggernaut. Far from being a fad, the service has grown in just two years from zero to tens of millions of devotees, who send as many as 400 million photos a day — and that number is growing. Investors have taken notice and poured $123 million into Snapchat, valuing the revenue-less company at an eye-popping $2 billion.

Why are investors and other grownups so into Snapchat? They see a company that isn’t merely improving the photo-sharing craze established by Instagram (acquired by Facebook in April 2012 for $1 billion) but creating an entirely new form of communication. Its signature feature allows users to share photos and videos that disappear forever after just a few seconds. (The sender chooses the expiration time, up to 10 seconds, using a clever timer in the Snapchat app.) Just as email revolutionized the way people “talk” to each other at work and Twitter (TWTR) is altering the way people broadcast information, Snapchat has in a very short time become the lingua franca of a younger generation.

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Snapchat’s ascent has caught its rivals off-guard — Facebook’s buyout offer (after it tried to copy it) suggests that the rest of the social media ecosystem covets what Snapchat has created. Google (GOOG) and Yahoo (YHOO) both are investing heavily in building out features that encourage users to interact with photos. Meanwhile, around the world, especially in China, similar “short burst” communication apps, such as Whatsapp, WeChat, Kik, MessageMe, KakaoTalk, and Line, are on the rise. They allow users to replicate the texting experience without the fees, a trend the phone companies must be eyeing nervously.

Snapchat’s future success is by no means assured. The ephemeral nature of transactions on the platform, the very thing that makes Snapchat so tantalizing to its audience, is a turnoff to marketers, who are looking for ways to monitor and track customers’ every move. Nevertheless, the Snapchat story, while still in its early chapters, offers important insights into the speed with which mobile services can proliferate. Inexperience and bravado, once traits that business executives sought to downplay (remember the “adult supervision” that investors used to impose on young founders?), are now assets, though financiers hedge their bets by providing founders with plenty of mentoring and support. If nothing else, Snapchat’s story serves as a reminder that innovations are always around the corner, and no company is immune to disruption — not even Snapchat itself.

In the spring of 2011, Evan Spiegel and Reggie Brown, both juniors at Stanford University, hatched an idea to build a mobile-phone app. It featured a unique approach: the disappearing photo. Spiegel and Brown’s own classmates dismissed the concept. When Spiegel presented it to his product-design class, his peers shot it down. Still, Spiegel and Brown believed they were on to something. They recruited Bobby Murphy, a 2010 graduate and a fellow member of the Kappa Sigma fraternity, with whom Spiegel had collaborated on an earlier, failed startup. (Called Future Freshman, a website for high schoolers, it never graduated.)

That summer the trio set to work at the home of Spiegel’s father, John, a prominent corporate litigator in Los Angeles. John Spiegel’s $3.3 million home in the swank Pacific Palisades neighborhood defied the startup narrative: There were no late nights spent wolfing down cheap takeout burritos à la Silicon Valley. These founders had access to a personal chef and a pool, and they indulged in L.A’.s nightlife scene. By mid-July they’d launched an app with a yellow “Ghostface Chillah” logo (an homage to the rapper Ghostface Killah). They originally called the service Picaboo — a mashup of “picture” and “peekaboo” — meant to convey the signature functionality of their idea: Now you see it, now you don’t.

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The friends moved quickly to turn the idea into a company. After just three months of coding, they launched a prototype in Apple’s App Store — and had an internal dispute so contentious it will surely require large sums of money and a gaggle of lawyers to resolve. After Spiegel and Murphy quarreled with Brown over the order of their names listed on Snapchat’s patent application, the two set the proverbial timer on Brown’s involvement with Snapchat and allegedly pushed him out of the company. Brown, now an MBA student at Duke University, is suing Snapchat, seeking damages plus a third of the company’s equity — suddenly a meaningful demand.

Meanwhile, in September 2011, Spiegel and Murphy renamed the app Snapchat, and it immediately began to catch on. Its first users were Orange County high school students who’d learned of it from one of Spiegel’s cousins. The students had been equipped with iPads, but Facebook was blocked. (Hey, it was school!) Snapchat was a great workaround. Within two months it had 3,000 users. Then Apple (AAPL) introduced a reverse button for the camera on its phone, designed to make selfies easy. Growth took off.

In many ways the timing was perfect for Snapchat’s explosion. The app caught the attention of a generation of kids smartening up to the dangerous potential of social media. The web had become a scary place, where every faux pas became permanent. Anthony Weiner had gotten caught with his pants down. Literally. College administrators began reviewing prospective students’ social media profiles before admitting them. New graduates learned that potential employers could view the tangible evidence of their youthful indiscretions. A racy communication tool that left no fingerprints suddenly looked attractive.

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By May 2012, Spiegel had taken a leave from his studies, and he and Murphy were running low on cash. (The pair had previously self-financed, relying on Murphy’s salary as an engineer with Revel Systems, an iPad point-of-sale company, to pay the server bills. In April 2012, Murphy had quit his job.) Meanwhile, venture capitalists were beginning to notice the app’s impressive growth. Lightspeed Venture Partners’ Jeremy Liew heard about it through a colleague, who had noticed the app on his teenage daughter’s phone. He tracked Spiegel down — via a Facebook message — to make the first investment. Spiegel and Murphy used the $485,000 from Lightspeed to hire engineers and pay for computing power. Later in the year Silicon Valley heavyweights Benchmark Capital and SV Angel joined Lightspeed to invest another $12.5 million.

While the service didn’t initially attract people over 25 years old, the sheer speed of its growth caused a small subset of power brokers to pay attention. Sony Pictures CEO Michael Lynton’s kids attended school at Crossroads, the same posh liberal prep school Spiegel had attended. Lynton’s wife emailed Spiegel to come to dinner, and soon afterward, in June 2013, the studio head joined the company’s board. Around the same time, Snapchat raised $60 million in a funding round led by IVP that valued the company at $800 million. As it turns out, that was nothing. Less than six months later hedge fund Coatue Management invested $50 million at a valuation of $2 billion.

Early on, the app caught the attention of Mark Zuckerberg, Facebook’s CEO. In December 2012 he flew to Southern California to meet with Spiegel and Murphy. Shortly after that, Facebook began experimenting with a copycat feature called “Poke,” named for a popular game-like feature from Facebook’s earliest days. It was a move that Spiegel referred to at a fall tech industry conference as “the greatest Christmas present we ever got” because of the attention Facebook’s assault focused on Snapchat. Poke failed.

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Outwardly Zuckerberg maintained that as social media matured, kids were doing more of it, so they could embrace lots of services without abandoning Facebook. “People assume that because there are different experiences, somehow it’s a zero-sum game,” he told Fortune in April. “But it’s actually a rapidly expanding market. People share more and more every year, and they want lots of different experiences.”

Despite the brave face Zuckerberg put on, Facebook was already concerned that Snapchat could steal some of its thunder with teenagers. Last fall it tried to buy Snapchat, offering to pay more than $3 billion, according to multiple sources close to both companies. The spurned offer was the first in a chain of Snapchat-related dings for Facebook. On Oct. 30, in an earnings call with investors, Facebook disclosed that daily active use among teens had fallen off. Then in early December Snapchat named a new chief operating officer: Emily White, the 35-year-old Facebook executive who had been charged with making that company’s Instagram unit profitable.

In October, Snapchat introduced “stories,” a new feature in which photos don’t immediately disappear. Users and their friends can build “chains” of photos, basically a digital photo album, that are available to all their contacts for 24 hours. (There’s also an option to share stories with all Snapchat users.) The community can view the album an unlimited number of times until it expires. Stories represents an evolution for Snapchat, offering customers slightly less privacy and urgency than the original product promised, while edging toward the kind of product marketers find appealing. Stories could be tweaked to include more contributors (including corporations) and to allow images to be accessible for longer periods.

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Snapchat says the new feature was under development for more than a year, and other moves suggest that the company has been getting ready for primetime. This fall it gave up the powder-blue beach bungalow it was using as headquarters for a much larger, less flashy storefront hidden in a Venice alley. There’s no signage anywhere; a tiny gray ghost painted on the door is the only indication that 30 or more employees are coding within. Earlier in the year it hired Philippe Browning, a mid-level mobile-advertising executive from CBS Interactive. His title — vice president of monetization — suggests a seriousness of purpose for Snapchat. It also has begun approaching well-known online marketers to encourage them to consider using Snapchat.

So far the response has been tepid. Only a few marketers are running tests on Snapchat. Taco Bell, for example, used Snapchat in May to distribute photos of its new Beefy Crunch Burrito, and it publicized its photo campaign on Twitter. The restaurant chain is now experimenting with stories. Some potential partners maintain that the service remains too new to be on marketers’ radar screens. “I’m on the boards of Chegg, Groupon, and Caesars,” says Jeff Housenbold, CEO of the market-leading photo site Shutterfly. “Snapchat is not entering into the marketing conversation at all. In contrast, Pinterest is entering the discussion all the time. But not Snapchat.”

Snapchat must persuade marketers that the benefits of reaching its coveted youth audience outweigh the relative anonymity of that audience. “In an era where big data rules, how does Snapchat, which collects no data on the user, sell itself to marketers?” asks Arra Yerganian, chief marketing officer for University of Phoenix, a big online advertiser.

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Marketing isn’t the only way Snapchat might make money, of course. Game publishers have used messaging platforms in Asia, including Chinese online giant Tencent’s WeChat, to publicize its games and allow users to share their experiences. Snapchat would work well that way. After building up a bigger user base, Snapchat also could experiment with premium services, such as storing photos.

With its $100 million-plus in venture money and a new operating chief, Snapchat certainly has the resources to try new products and find ways to court companies that will pay for access to the Snapchat universe. But given the speed of innovation going on in China and Silicon Valley (and now Los Angeles), what Snapchat does not have is unlimited time. And a company with a timer as part of its app is sure to realize that.

Reporter associates: Marty Jones and Chanelle Bessette

This story is from the January 13, 2014 issue of Fortune.