Bitcoin seeks to be an electronic cash (currency) system that doesn't rely on trust. Paradoxically, Bitcoin requires a trust-based ecosystem.
As a brief summary: The Bitcoin system was developed as an electronic currency by Satoshi Nakamoto (apparently, a pseudonym). Bitcoins exist only in the online world (they have no physical form). Each Bitcoin is uniquely identified, and is part of a limited edition (only a pre-set number will be issued). And, if properly executed, Bitcoin transactions are anonymous and non-reversible. For a more detailed explanation of Bitcoin's architecture, see Benjamin Wallace (Wired) or The Economist.
Bitcoin is intended to be digital cash/currency, based on cryptography and peer-to-peer networks, rather than trust. As Nakamoto explains:
"The root problem with conventional currency is all the trust that's required to make it work. The central bank must be trusted not to debase the currency, but the history of fiat currencies is full of breaches of that trust. Banks must be trusted to hold our money and transfer it electronically, but they lend it out in waves of credit bubbles with barely a fraction in reserve. We have to trust them with our privacy, trust them not to let identity thieves drain our accounts."