Bitcoin is basically a digital currency or online currency or virtual currency: a form of money stored in an owner’s online “Virtual Wallet”. It is the first decentralized digital currency i.e., free from the control of governments or central banks.
Bitcoin is the most successful crypto-currency to be created. Crypto-currency is an attempt to replace money transactions with a digital medium of exchange using peer-to-peer networking technology. The idea is that you use cryptography to control the creation and transfer of money, rather than relying on central authorities.
Bitcoin was released in 2009 by a mystery developer, Satoshi Nakamoto who wanted a virtual currency that was devoid of regulation. It was first suggested in 1998 by Wei Dai on the cypherpunks mailing list and it was built around a philosophy that money is any object accepted as payment.
Since the success of Bitcoin, there have been others which are starting to make names for themselves such as Ripple, Litecoin, Peercoin, Mastercoin, Namecoin and Quarkcoin. Even publications like Arstechnica and local councils in the UK like Hull, have come up with their own crypto-currencies.
Many other crypto-currencies have just died because no one used them. Non-Bitcoin crypto currencies are collectively known as altcoins and they are more or less based on the same idea.
Their success depends on how much "cash" (the total value of transactions) they have running around the peer-to-peer network (i.e., the virtual economy). Since Bitcoin is open-source, anyone can develop their own crypto-currency using the same technology.
Bitcoins derive their value partly through their scarcity, which is defined by a cryptographic lottery. You can buy Bitcoins on online crypto currency exchanges or you can earn them through "mining."
Bitcoin mining programs compute an encryption function called a "hash" on a set of random numbers. Coins are awarded every 10 minutes to whichever miner happens to compute a number below a certain threshold.
A few years ago, Bitcoin mining was handled by standard PCs with powerful graphics cards, but as the hash difficulty has gone up, the only way to mine Bitcoins is to employ a Bitcoin ASIC, a chip that has been designed specifically for this task.
This lottery favours those with the biggest and fastest machines and by 2014, there will be about 12 million Bitcoins in circulation. Note that the total number of Bitcoins in (virtual) circulation will never exceed 21 million because of the way the system was designed.
As the Bitcoin network gets bigger, the hash gets more complex and miners, they get fewer Bitcoins for their trouble, and hence they always need better hardware and higher Bitcoin prices to make it worthwhile.
The running of this lottery is what powers the financial transactions behind the network. In theory, it means that it is highly secure, untraceable, fast, and practically free.