From the user's perspective, Bitcoin is a mobile app or computer program that provides a personal Bitcoin wallet that users can use to send and receive Bitcoins. This is how Bitcoin works for most users.
Behind the scenes, the entire Bitcoin network shares a public ledger called the "blockchain". This ledger contains every transaction ever processed, allowing users' computers to verify the validity of each transaction. The authenticity of each transaction is protected by a digital signature corresponding to the sending address, which allows users to have full control over the Bitcoins transferred from their own Bitcoin addresses. In addition, anyone can use the computing power of specialized hardware to process transactions and be rewarded with Bitcoins for doing so. This service is often called "mining". You can read more about Bitcoin on the dedicated page and the original paper.