In short, you convert your actual money into a digital currency, and can then transfer it using your smartphone or computer to transfer the funds. Bitcoin does not use a bank and depends on a peer-to-peer network.
From a Washington Post article last month:
The coin in question now has a global circulation worth more than $1.4 billion on paper. Yet almost no one, it seems, knows the true identity of its creator. In the United States, this mysterious money has become the darling of antigovernment libertarians and computer wizards prospecting in the virtual mines of cyberspace. In Europe, meanwhile, it has found its niche as the coinage of anarchic youth.And, from Lifehacker:
You can obtain Bitcoin's either by trading other money, goods, or services with people who have them or through mining. The mining process involves running software that performs complex mathematical equations for which you're rewarded a very small portion of a Bitcoin. When you actually have some of the currency, you can then use it to purchase anything that accepts it.The anonymous transfer of money can be good for privacy, but authorities are worried about its use for money laundering and the buying and selling of illegal items. ("The FBI, for example, says that 'the way it creates, operates and distributes bitcoins makes it distinctively susceptible to illicit money transfers.'")
Finally, the last word from a different Washington Post article:
But Bitcoin still makes a lot of people uncomfortable. Regulators and law enforcement officials by and large view Bitcoin as a dangerous tool for money laundering. Others refer to Bitcoin as an underground banking system or the currency of those who seek to engage in more controversial activities — such as financing the development of 3D-printed guns. And Bitcoin may not be as safe as its supporters would like to think, no matter how strong the encryption, given the Bitcoin heists and hacks of the past 18 months.
The rise of the bitcoin: Virtual gold or cyber-bubble?, The Washington Post
Trojan Turns Your PC Into Bitcoin Mining Slave, Wired
Bitcoin And The End Of Money, TechCrunch What Is Bitcoin and What Can I Do With It?, Lifehacker Man Wants To Be The First To Sell A House For Bitcoins Because Real Money Is So 2008, Consumerist
What Is A Bitcoin?, Washington Post
Why reports of Bitcoin’s death may have been greatly exaggerated, The Washington Post