Square, which made it possible for the smallest, most accidental merchant to accept credit cards by eliminating expensive equipment and high fees, has a new idea: Banish the cash register. And paper receipts, while we’re at it.

On Monday, Twitter co-founder Jack Dorsey unveiled Square Register, which will allow merchants to manage inventory and run analytics against their sales — without expensive equipment and high fees.

Only about 50 vendors are launch partners for Register, which is accompanied on the customer side by a sort of virtual card case that makes it easier to pay at Register establishments and is designed to look like a wallet containing loyalty cards.

But despite the softish arrival for this enhancement, Square itself seems on solid footing: Dorsey said that the company has shipped 500,000 Square readers, and on Sunday Dorsey Tweeted that the company had booked $3 million in a single day for the first time. Square reached the $2 million milestone less than a month earlier. It took 15 months to hit the $1 million mark.

Dorsey’s latest project represents just one of several new developments in the rapidly evolving digital payments space. Last week, financial services giant Visa announced plans for a digital wallet based on Near Field Communication, or NFC, technology, which goes Square one better by not even requiring the use of the small dongle that fits into the earphone jack.

NFC-devotees believe the technology will become the standard format for mobile payments. NFC allows the secure, short-range communications needed to make mobile point-of-sale payments. Google reportedly plans to begin testing its own mobile-payments system within the next few months, and will install thousands of NFC units made by VeriFone Systems at merchant locations in New York and San Francisco.

Only a handful of phones currently support NFC, including Google’s Android-based Nexus S, which is produced by Samsung. There had been some chatter that Apple would incorporate NFC into the next-generation iPhone 5, but Bernstein Research analyst Toni Sacconaghi recently cast some doubt on that notion, in part because the NFC infrastructure rollout remains in its infancy. But there is little doubt that Apple is experimenting with NFC, as Sarah Clark of SJB Research and Near Field Communications World, told Wired.com recently.

Mobile payments are poised to explode. This year, internet-payment giant PayPal expects to process $2 billion in such payments — nearly three times the amount it processed in 2010, company spokesperson Sara Gorman told Wired.com recently.

Dorsey’s dramatic Square Register launch will only further fuel the all-out race for advantage in the mobile payments space. In the current issue of Wired magazine, Dorsey explained his motivation.

Money is a concept that’s been with us for 5,000 years, and it’s never been designed to be anything but a burden. You come into my coffee store and order a cappuccino, and I hit the Cappuccino button on the cash register and see that it’s $3.24. And I take your credit card and type “$3.24? into the credit card terminal. I swipe your card and give you the credit card receipt to sign. Then I take that back and staple it to the cash register receipt and give it back to you, along with your credit card. Then you take back the credit card and throw the receipts away. And meanwhile, at the end of the day I have no idea how many cappuccinos I sold because it’s really difficult to access that information. It’s completely useless.