Bitcoin is a decentralized digital currency that can be sent anonymously via the internet to anyone without the need of a bank account. All you need is a computer, an internet connection and a Bitcoin wallet.
When Bitcoins are transferred from a wallet to another wallet, an electronic signature is automatically added and the transaction broadcasted over the internet. Shortly after it is verified by a network of "miners" and permanently stored in the Bitcoin network, proving that those Bitcoins now belong to the receivers address.
Small amounts of Bitcoins are generated all over the world by people running this free software called a "Bitcoin miner" on their computer. These miners verify transactions by competing against each other to solve complex algorithms in exchange for a small reward. The Bitcoin network self-adjust according to the combined mining power ensuring that just a limited amount of Bitcoins are generated at a predictable rate.
Mining with a normal computer that has a standard graphic card such as "ATI Radeon 7970" will generate about 0.02 Bitcoins worth $3.31 every 24hours, with today's difficulty rate (23.05.13)
Bitcoins has several great advantages over government backed money, the fees are minute and its possible to send Bitcoins anywhere in the world within seconds. Your funds can not be frozen and there is a finite amount of 21 million Bitcoins that can ever exist.