Bitcoin is a decentralized digital currency with no controlling institution or government. Bitcoin uses a peer to peer (p2p) network to coordinate a worldwide, universal ledger where all transactions on the bitcoin network are recorded. This ledger is called the blockchain, and is the core of Bitcoin's success.
When someone makes a bitcoin transaction, it is verified and broadcast across the entire network. Depending on the transaction fee (which the sender decides - a higher fee verifies faster) the transaction may take less than 10 minutes to verified, and put into what is called a block.
The bitcoin network consists of many computers attempting to "mine" blocks - each time a block is found, all pending transactions on the network are sealed within it on the blockchain, and they are considered verified. This bitcoin mining process essentially involves computers making millions of guesses as to the block solution, which is designed to take roughly 10 minutes regardless of how many computers are mining bitcoin. The mining process is also the only way new bitcoins are brought into existence - each block found has a reward value, that decreases every 4 years until eventually it will reach zero.
Bitcoin transactions are mostly anonymous - because the blockchain is public, all transactions are public knowledge to anyone. However, bitcoin wallets don't have names registered to them, so only the transactions and values are public. Linking a bitcoin address to a person is not so easy. They're also irreversible - once a bitcoin transaction has been verified, there's no going back. There's no credit card chargeback option or central authority to raise a dispute with.
Bitcoin has only been around a few years, yet it’s growth and resilience seem to indicate that the digital world we live in requires a new kind of money.
You can go to bitcoin.org to learn more and get started with Bitcoin.