How does Bitcoin work?
This is a question often surrounded by confusion, so here's a quick explanation!
The basics for a new user
As a new user, you can get started with Bitcoin without understanding the technical details. Once you've installed a Bitcoin wallet on your computer or mobile phone, it will generate your first Bitcoin address and you can create more whenever you need one. You can disclose your addresses to your friends so that they can pay you or vice versa. In fact, this is pretty similar to how email works, except that Bitcoin addresses should be used only once.
Balances - blockchain
The blockchain is a shared public ledger that the entire Bitcoin network relies on. All confirmed transactions are recorded in it, which lets wallets calculate their spendable balance and lets the network verify that the sender is authorized to spend those coins. The integrity and chronological order of the blockchain are protected by cryptography.
Transactions - private keys
A transaction is a transfer of value between Bitcoin wallets that gets recorded in the blockchain. Bitcoin wallets keep a secret piece of data called a private key, which is used to sign transactions, providing a mathematical proof that they came from the owner of the wallet. The signature also prevents the transaction from being altered by anybody after it has been issued. All transactions are broadcast to the network and usually receive their first confirmation within about 10 to 60 minutes, through a process called mining.
Processing - mining
Mining is a distributed consensus system that is used to confirm pending transactions by including them in the blockchain. It enforces a chronological order, protects the neutrality of the network, and allows different computers to agree on the state of the system. To be confirmed, transactions must be packed into a block that follows very strict cryptographic rules verified by the network. These rules prevent earlier blocks from being changed, because doing so would invalidate every block that comes after. Mining also works like a competitive lottery that makes it difficult for any single participant to keep adding new blocks one after another. This is what keeps any group or individual from controlling what goes into the blockchain, or rewriting past parts of it to reverse their own spending.
Going down the rabbit hole
This is just a short summary of Bitcoin. If you want to learn more of the details, you can read the original paper that describes its design, the developer documentation, or explore the Bitcoin wiki.